a
OU 166603 >m
OSMANIA UNIVERSITY LIBRARY
CallNo. pf / * * /\*\ fJL.- ^^^sion No. /(f X?
Author
This book should be returned on or before the date last marked belo
INTRODUCTION TO STATISTICAL MECHANICS
FOR STUDENTS OF PHYSICS AND
PHYSICAL CHEMISTRY
PUBLISHED BY
Constable & Company Limited
London W.C.2
BOMBAY
CALCUTTA MADRAS
LEIPZIG
Oxford, University
Press
TORONTO
The Macmillan Company
of Canada, Limited
INTRODUCTION TO STATISTICAL
MECHANICS'^ STUDENTS
OF PHYSICS AND PHYSICAL
CHEMISTRY
BY
JAMES EJUJi, M.A.,
ASSOCIATE PROFESSOR IN THE DEPARTMENT OF
PHYSICS, UNIVERSITY OF LIVERPOOL.
WITH A FOREWORD BY
F. 0. DONNAN, C.B.E., M.A., Ph.D., D.Sc., F.B.S.
PROFESSOR OK CHEMISTRY, UNIVERSITY COLLEGE, LONDON
LONDON
CONSTABLE & COMPANY LTD
1930
First published 1910
PRINTED IN GREAT BRITAIN BY THE WH1TEFRIARS PRESS, LTD.
LONDON AND TOJNBRIDGK.
FOREWORD
PHYSICO-CHEMICAL science regards the spatial universe as
filled with a vast multitude of moving " elements " which
possess both parfciculate and wave -like characters the
" wavicles " of Eddington. The only way to render this
elusive and protean microcosmos amenable to mathematical
calculation, and to interpret physical measurement, is to
deal in terms of probabilities, to employ the statistical
method. The macroscopic world of sense and the measure-
ments based thereon possess the validity characteristic of
the averages of an actuarial estimate. The statistical method
is therefore of profound importance. It dominates the
whole of modern science. Combined with the generalised
principles of dynamics and the quantum theory it has
produced statistical mechanics and " quantum statistics."
These are not special branches of science peculiar to the needs
of a few lonely mathematicians. The simple truth is that
they constitute the fundamental basis of modern physical
science. But another simple truth, though a painful one, is
that they involve a severe discipline for the untutored human
mind.
Fortunately, Professor Rice has now come to tutor our
minds and bring consolation to our hearts. This book of
his is a first-rate one, for which all serious students of
chemistry and physics will owe him a deep and lasting debt of
gratitude. He has explained and expounded the principles
of statistical mechanics and quantum statistics with extreme
lucidity. In the earlier portions of the book the general
principles of probability and statistics are developed and
applied to the solution of many important problems. Then
the concepts of the quantum theory are introduced, and
finally the generalised principles of dynamics. The most
recent advances associated with the names of Bose, Einstein,
Fermi and Dirac are dealt with in an important appendix.
In excellent appendices to many of the chapters the author
vi FOREWORD
succeeds admirably in removing the mathematical difficul-
ties inherent in parts of the reasoning. I would particularly
and very warmly commend this book to the attention of
students of physical chemistry it needs no recommendation
to students of physics. Subjects such as chemical equilibria
in gas reactions, the specific heats of gases and solids, the
entropy of a perfect gas, the Nernst heat theorem, the
chemical constants, the theory of the atom, the Einstciri-
Smoluchowsky theory of density fluctuations and collision-
form ula? arid chemical kinetics are all fully and clearly
explained. Indeed, I will without hesitation make the two
following assertions : (1) every student of physical chemistry
must read the book ; (2) no student of physical chemistry
familiar with the calculus will experience any serious diffi-
culty in mastering its contents.
I make these assertions because I believe this book is
destined to exert a great influence on the training of the
present generation of chemists and physicists. As the title
indicates, Professor Rice has had in view and for that we
cannot be too profoundly thankful the needs of chemists
as well as physicists. He has certainly succeeded in his
object. But he has done so without any slurring over or
evasion of difficulties. Throughout the work the treatment
is thorough and complete. I would specially commend to
the attention of chemists (and physicists) the two chapters
which deal in a most clear and original manner with the
second law of thermodynamics. The light which this statis-
tical analysis of the microcosmos throws on one of the greatest
if not the greatest of the experiential generalisations
of the macrooosmos constitutes, to my mind, one of the great
triumphs of the human mind. Seldom, if ever, has it been set
forth with such masterly lucidity and logic.
This book is clearly the fruit of many years of study and
thought. I wish it many years of prosperous and beneficent
influence.
F. G. DONNAN.
THE SIR WILLIAM RAMSAY LABORATORIES OF
PHYSICAL AND INORGANIC CHEMISTRY,
UNIVERSITY COLLEGE, LONDON.
PREFACE
THE aim and scope of this book is indicated in the intro-
ductory chapter. The author is not blind to the fact that
the student he had in mind when he wrote it is not going to
read some parts of it without a serious mental effort ; the
necessarily mathematical form of the arguments entails that
result. But he feels certain that any student with the
average mathematical equipment acquired in the first two
years of a University science course will not find it impossible
to follow the details of the treatment, and as much assistance
as possible is provided in explanatory appendices, in verbal
interpretation and illustration. In the proofs no steps are-
omitted. On that account the content of subjects dealt
with has had to be restricted ; otherwise the book would
have grown to a length unsuitable for the type of reader
whom it is intended to assist. The problems treated are con-
cerned with systems in statistical equilibrium, although a
short appendix refers to the subject of collision-frequency in
gases and its bearing on chemical reactions.
The author's best thanks are due to Dr. A. McKeown
and Mr. E. A. Stewardson, both of Liverpool University, for
helpful advice and criticism, and to the former for reading
the proofs.
J. RICE.
UNIVERSITY OF LIVERPOOL,
October, 1929.
Tii
TABLE OF CONTENTS
HAPTER PAGE
FOREWORD ....... v
PREFACE ....... vii
INTRODUCTION ...... 1
I. STATISTICAL METHOD ..... 6
APPENDIX. THE NORMAL LAW OF ERRORS . 14
II. THE STATISTICS OF A SIMPLE MOLECULAR SYSTEM 21
APPENDIX A. STIRLING'S THEOREM . . 28
APPENDIX B. PREPONDERANCE OF CERTAIN
STATES ....... 29
III. THE PROBABILITIES OF THE DIFFERENT STATES
OF A SIMPLE MOLECULAR SYSTEM ... 34
IV. TEMPERATURE AND THE DISTRIBUTION CONSTANT 45
V. EXTENSION TO MORE COMPLEX MOLECULES . 57
VI. THE SECOND LAW OF THERMODYNAMICS . . 68
VII. THE ENTROPY OF A PERFECT GAS ... 77
VIII. THE STATISTICAL THEORY OF CHEMICAL EQUI-
LIBRIUM IN A GAS REACTION ... 82
IX. INTERMOLECULAR FORCES .... 92
X. FLUCTUATIONS OF DENSITY IN A MOLECULAR
SYSTEM 102
XI. THE SECOND LAW OF THERMODYNAMICS II . 114
XII. THE STATISTICAL-MECHANICAL THEORY OF A
LIQUID AND A VAPOUR PHASE IN CONTACT . 118
XIII. THE SOLID STATE CONSIDERED AS A SIMPLE
LATTICE OF MASSIVE PARTICLES . . .123
XIV. THE QUANTUM HYPOTHESIS . . . .126
XV. THE THEORY OF THE STATIONARY STATES OF AN
ATOM 143
XVI. DISTRIBUTION OF A SYSTEM IN ENERGY . . 151
APPENDIX. DEGENERACY . . . .161
S.M. ix b
x TABLE OF CONTENTS
CHAPTER PAGE
XVII. QUANTUM THEORY OF THE SPECIFIC HEATS OF
GASES 164
XVIII. THE ELASTIC SPECTRUM OF A LINEAR LATTICE
OF COHERING PARTICLES . . . .174
XIX. THE ELASTIC SPECTRUM OF A CUBICAL LATTICE . 189
XX. THE SPECIFIC HEATS OF SOLID BODIES . . 207
XXI. THE ENTROPY CONSTANT OF A GAS . . 213
XXII. THE ENTROPY CONSTANT OF A MON ATOMIC GAS
AND STATISTICAL MECHANICS . . . 221
XXIII. ENSEMBLES OF SYSTEMS I .... 234
XXIV. ENSEMBLES OF SYSTEMS II .... 246
APPENDIX ON RECENT DEVELOPMENTS
I. BOSE'S STATISTICS OF LIGHT QUANTA IN A TEM-
PERATURE-ENCLOSURE 270
II. EINSTEIN'S THEORY OF AN IDEAL GAS . . 275
III. THE FERMI-DIRAC STATISTICS .... 277
IV. THE STATISTICAL METHOD OF DARWIN AND
FOWLER 282
APPENDIX ON COLLISION FORMULAE AND
CHEMICAL KINETICS
I. COLLISIONS BETWEEN MOLECULES IN A GAS . 296
II. COLLISION-FREQUENCY AND EQUILIBRIUM. THE
H THEOREM 306
III. THE KINETICS OF GAS REACTIONS IN A HOMO-
GENEOUS SYSTEM 311
Note on Chapter X. THE EINSTEIN FLUCTUATION
FORMULA ....... 327
SUGGESTIONS FOR FURTHER READING . . 321)
INDEX 331
INTRODUCTION TO
STATISTICAL MECHANICS FOR
STUDENTS OF PHYSICS AND
PHYSICAL CHEMISTRY
INTRODUCTION
THE experimental data which have acted as a guide to the
discovery of the laws of physics and chemistry have been in
the main derived from careful observation of the behaviour
of very limited portions of matter deliberately and skilfully
placed by the experimenter in artificial surroundings. The
purpose of such an environment is the elimination as far as
may be possible of all those external influences which under
more normal circumstances would affect the behaviour of
the portion of matter considered, but which are of no imme-
diate interest to the observer, and only interfere with his
search for the effect of some special influence with which he
is at the moment directly concerned, and which is allowed by
the special circumstances of the experiment to have full
play.
Such a portion of matter so situated may be termed a
" system," and although its material constitution may not
be simple, yet, by a strict limitation of the number of external
influences operating on it and by a restriction of the number
of properties observed, it may be possible to regard it as a
system of a simple nature. By the removal of artificial
limitations and the direction of the attention to a wider circle
of properties the system becomes more and more complex.
Thus a small quantity of dry air, confined in a glass tube
above mercury, whose temperature is maintained constant,
is an example of an extremely simple system whose change
of volume under change of external pressure is the sole point
2 STATISTICAL MECHANICS FOR STUDENTS
of interest for the observer. The system would be equally
simple if the pressure were maintained constant and observa-
tion directed towards change of volume caused by change of
temperature. The system becomes more complex if both
pressure and temperature are allowed to vary, but the
information already obtained from the simple cases enables
us to predict (with success as it so happens in this case) what
happens under the wider operation of external influences.
If we now consider a portion of matter consisting of air and
water, the system, now having two distinct phases (an air-
vapour phase and a liquid phase), is more complex, inasmuch
as we naturally observe two quantities, viz., the volume of
each phase, when conditions of pressure and temperature
are varied. Information derivable from the study of such
a system and of others still more complex, is applied by the
meteorologist to large tracts of our atmosphere or may even
be of service in general considerations concerning the
atmosphere as a whole regarded as a single but very complex
system.
Systems are sometimes defined as "physical " or " chemi-
cal " according as the observations made are related to
change in physical properties or in chemical constitution,
but the terms are not always definite and no clear separa-
tion into two classes is in general possible. We shall use
the term " physical system " to embrace all systems in which
chemical as well as physical properties are observed and to
which the epithet " physico-chemical " might be attached
save for its clumsiness as a word.
Historically, the systems^hich were the first to be treated
successfully by the methods of exact observation and
mathematical analysis initiated in the sixteenth and seven-
teenth centuries, are the mechanical devices, bodies moving
on or near the earth's surface and the system of planets and
satellites attendant on the sun. Here the observed proper-
ties are the relative positions of the various bodies or parts
of the system and the changes produced by the forces acting
mutually between the various parts or exerted on them by
bodies external to the system. The famous laws of motion
propounded by Newton as an adequate summary of the
INTRODUCTION 3
experimental facts, and applied by him with wonderful skill
and success to a wider range of such phenomena, were
extended and provided with a very complete mathematical
formulation by D'Alembert, Lagrange, Laplace, Hamilton
and others. Such formulations became known as various
analytical ways of stating the " Principles of Dynamics/' and
a system of bodies in which the movement of its parts
(assumed to conform exactly with these principles) is the
prime object of observation is termed a " dynamical system."
However, motion is only one of the observable features of
any collection of bodies. Properties such as temperature,
pressure, quantity of heat, luminosity, colour, refractivity,
magnetic induction, electric charge and potential, chemical
constitution, reactivity, etc., claim our attention. For some
time several of these properties were explained by postulating
the existence of subtle and weightless forms of matter, not
accessible to direct observation, such as " caloric," " mag-
netic and electric fluids." But the influence of the mathe-
matical physics of the eighteenth century, with its treatment
of the movement of a finite body as arising from the
interaction of the discrete " particles " constituting the body
and from their response to external forces, gave an irre-
sistible impulse towards the explanation of all physical and
chemical properties of matter as manifestations of the
configuration and motion of the ultimate particles of the
matter. It was not an accidental circumstance, but one
quite natural in the mental environment of the time, that
Dalton should have been led to the formulation of his atomic
hypothesis by considerations of a mathematical-physical
rather than of a purely chemical nature. There followed in
quick succession the dynamical theory of heat, the kinetic
theory of gases, the molecular theory of magnetism, all
meeting with stubborn resistance but all winning recognition
by their power in summarising experience and by the
ultimate identity of the underlying ideas in each case.
Finally, as the pinnacle of this edifice, built on a dynamical
view of all properties of matter, there was constructed the
theory that radiation is ultimately a propagation of an
actual wave motion in a medium possessing elastic and
B 2
4 STATISTICAL MECHANICS FOR STUDENTS
inertial properties like solid matter, those properties them-
selves being thrown back in the writings of Cauchy and
others on the interaction of ultimate ether particles far sur-
passing material atoms in minuteness and fineness of struc-
ture. Even biological science could not evade the influence
of these ideas, and in the nineteenth century there was for a
time a great vogue in the idea that in some way life and
consciousness are but the by-products of the mechanical
reactions between the ultimate atoms of living tissue, whose
movements are just as much determined by the laws of
dynamics as are those of the planets in our solar system.
To be sure such crude notions have had their day in biology,
and even in physical science, " atoms interacting across
empty space " and " waves pulsating through the lumini-
ferous ether " are strongly suspect. But although the
conceptual entities which we employ in order to give our
minds an orderly picture of the apparently inextricable
complexity of natural phenomena are being replaced at the
moment by new and as yet unfamiliar concepts, there still
remains as powerful as ever and absolutely indispensable
the great body of mathematical analysis which has grown
up with the physical science of the past two or three cen-
turies ; and in that body analytical dynamics holds a
fundamental place for the reasons already stated.
This mechanical conception of the underlying nature of
physical phenomena is familiar enough to anyone conversant
with the usual texts of Physics and Chemistry. The notions
are entirely plausible in a qualitative or roughly quantitative
sense. It is when one goes into the matter in some detail
and attempts to apply mathematical methods in order to
produce quantitatively precise or nearly precise results that
trouble begins. Molecular systems are much too complex
to follow in detail with the aid of dynamical laws and hypo-
theses as to the nature of intermolecular forces or intra-
molecular and intermolecular electromagnetic fields. To
make any headway at all the laws of probability have to be
impressed into service and made to co-operate with dynami-
cal principles. The worker is involved at once in statistical
considerations, and this combination of statistical calcula-
INTRODUCTION 5
tions with dynamical reasoning is called " Statistical
Mechanics." Fortunately, for those not too conversant
with dynamical methods, considerable progress in this
subject can be made towards tangible results without any
greater knowledge of mechanics than that possessed by the
average student at the end of a University second year in
the Applied Mathematics class room. At a pinch one can
manage along for a time on even less. In this book every
effort is made to keep at first to illustrations of such a nature
that a detailed knowledge of dynamical methods, such as
those employing the Lagrange and Hamilton equations of
motion, is not required. For a satisfactory foundation,
however, of the postulates upon which we base the statistical
calculations, a knowledge of Hamilton's equations is required.
Still, we shall assume that the postulates, explained at the
outset, are all right, and make use of them at once, deferring
their justification to the last chapters of the book. This
appeals to the author as being probably the manner of
laying out the work, which will evoke the interest of the
reader at once. As regards pure statistics, little more is
needed than an elementary knowledge of permutations,
such as is available in any algebra text, and of the binomial
and multinomial theorems. It is assumed, of course, that
the reader has some knowledge of the symbolism and methods
of the calculus. Any special mathematical information
beyond this is supplied in appendices.
CHAPTER I
STATISTICAL METHOD
1 . 1 The Spin of a Coin. It is a commonplace statement
that on spinning a penny the chances are equal that it will
present a head or tail. The d priori probability of either is
0-5. The use of the epithet " a priori " might lead us to
infer that this is a statement deduced from our " inner
consciousness " or some equally mysterious source. Not
so ; it is a bald statement of the experimental fact that if
anyone chooses to amuse himself for some time by tossing
a coin repeatedly and at random, he will find the ratio of
heads to tails always close to unity, and the more so the
longer he proceeds with the entertainment. There are two
aspects presented by the fallen coin, and one has as good a
chance of showing itself as the other.
Suppose we spin two coins. How many " complexions "
are possible ? There are four, since the two pennies may
both present heads or both tails, or the first penny may
present a head and the second a tail, or vice versd. Any one
of these is as probable as any other, since the events are
independent ; for the fall of one coin (say) head up, does not
bias the fate of the other. Thus the a priori probability of
each of the four complexions is 0-25, and any " doubting
Thomas " can overcome his scepticism by trying it. He
will find that on making a large number of throws
practically one quarter of them will yield any given com-
plexion.
There are four complexions, but three " statistical
states " : (1) both pennies showing heads, (2) both show-
ing tails, (3) one showing a head and one a tail. Two
complexions fall within state (3), and so the probability of
that state is 0-5, while that of the states (1) and (2) are 0-25
each. State (3) is twice as probable as either (1) or (2).
STATISTICAL METHOD 7
Increase the number of coins thrown to three. There are
eight complexions. Here they are :
First Penny. Second Penny. Third Penny.
(1) ... Head ... Head ... Head
(2) ... Head ... Head ... Tail
(3) ... Head ... Tail ... Head
(4) ... Head ... Tail ... Tail
(5) ... Tail ... Head ... Head
(6) ... Tail ... Head ... Tail
(7) ... Tail ... Tail ... Head
(8) ... Tail ... Tail ... Tail
All are equally probable, having ^ as their a priori proba-
bility. There are four statistical states, viz. :
(1) All heads.
(2) All tails.
(3) Two heads and one tail.
(4) One head and two tails.
Only one complexion falls within either state (1) or (2),
but three within either of the third and fourth states.
Hence the states (3) and (4) are each thrice as probable as
(1) or (2). The probabilities are J, , f , f .
It should require little thought now to extend the reason-
ing to any number of coins. Let there be n coins, and
suppose we indicate the fact that the r th coin presents a head
by the symbol a r , and that it presents a tail by the symbol b r .
Then any complexion presented by a fall of the n coins will
be represented by some such expression as
aj a 2 6 3 a 4 6 5 a n .
Each term will consist of n symbols. In every term, the
suffixes will proceed regularly from 1 to n, but the arrange-
ment of the a and b symbols will be fortuitous. Thus the
term written represents a complexion in which the first
penny falls head up, the second head up, the third tail up,
the fourth head up, the fifth tail up, etc., the last head up.
We can find all the possible complexions by working out the
product of the n factors
(l + &l) (2+^ 2 ) (8+*s) K + *n) (1.1-1)
There are 2 n terms ; in each of them any suffix can only
8 STATISTICAL MECHANICS FOR STUDENTS
occur once as no penny can show both a head and a t^il in
one fall. The b priori probability of each complexion is
therefore 2~ n . How many statistical states are there and
what is the probability of each ? A state, e.g., in which r
pennies show heads and s show tails (r + s = n) will include
all those complexions whose symbolic terms contain r of the
a symbols and s of the 6 symbols. If we wish to find the
number of these complexions we obliterate the suffixes in
(1.1.1) and consider the coefficient of a r b* in the product
(a + b) n . By the binomial theorem this is
n !
TIT!
Each of the complexions has an a priori probability 2~ w .
Hence the probability of the statistical state mentioned is
r \s\ \2J
There are, of course, as many statistical states as there
are terms in the expansion of (a -f b) n , i.e., n + 1.
The reader should bear in mind the experimental basis
of these calculations. The results might have been other-
wise. If by some edict of the Master of the Mint pennies
were so loaded that each one fell twice as often head up as
tail up, the d priori probability of a head would be 2/3, of a
tail, 1/3. To take account of the increased chance of a head
being shown by any coin we must in the reckoning of com-
plexions consider the product
(2a x + 6 X ) (2a 2 + 6 2 ) (2a n + b n ).
The coefficient of any term will give the number of times
which the corresponding complexion will turn up on the
average out of 3 n tosses. Thus a complexion in which r
particular pennies jburn up heads and the remaining s tails
will have an b priori probability of (2/3) r (l/3)< or 2 r /3 n . So
the probability of the statistical state, r heads and s tails
would no longer be the coefficient of a r b 9 in
STATISTICAL METHOD
but in
In general, if the a priori probabilities of a head and a tail
were respectively # and q (p + q = 1), the probability of the
statistical state mentioned would be the coefficient of
a r b' in
(pa + q b)*,
i.e.,
FTTi^ * ' ' ' (1 ' 1 ' 3)
1 . 2 Throwing o! Dice. If we were to use dice instead of
coins, we should have six possibilities with each die, not
merely two as in the case of coins. Each aspect of a die
has an a priori probability 1/6, if the dice are not loaded.
Let a throw of one by the r ih die be symbolised by a lf , of
two by a 2r , etc., of six by a 6f . We can symbolise a given
complexion in which, say, the first die throws a three, the
second die a five, the third a five, the fourth a one, the fifth
a two, the sixth a four, the seventh a three, etc. the n ih a
two by the expression
In any such symbolic term the second suffix advances regu-
larly from 1 to n, but the first is fortuitously chosen from 1
to 6. Any possible complexion is represented by some one
of the 6 n terms obtained by expanding the product of the
n factors.
fall + 021 + + a 6l) (012 + <*22 + + #62)
...... (*+* + . + 0eJ (1-2.1)
Each complexion has an a priori probability of 6~ w .
We can find the number of different complexions within
the statistical state, in which n l dice throw a one, n 2 dice
throw a two, etc., 7^ 6 dice throw a six, by eradicating the
second suffix in (1 . 2 . 1) and calculating the coefficient of the
term.
a s n
10 STATISTICAL MECHANICS FOR STUDENTS
in the expansion of
(i + 2 + a 3 + a * + a s + a e) n
In any text-book of algebra the reader will find in the
chapter on the multinomial theorem that this is
n\
Thus the probability of this statistical state is
n\
\ n \ n I n \ n \ n ! V6
(1.2.2)
If the dice had been loaded alike, so that the a priori prob-
ability of a throw of a one by any die were p l9 of a two, p 2 ,
etc., (#! + p% + Pz + Pt + Pz + P Q ~ I)> the probability
of the state mentioned would be the coefficient of a/ 1 a 2 w *
a 3 n a 4 w * a 6 w ' a 6 n in
i.e. 9
V f
IV 9 p *i p *t p * p n* p * p * f (1.2.3)
Wj! n 2 \ %! n 4 ! n 5 \ n^l ' l 2 3 4 5 6
The extension of these results is now an easy matter.
Let there be n similar articles each one of which can present
at one time one of c different aspects, the a priori probability
of each aspect being p l9 p 29 p& , p c respectively, then
the probability of the statistical state in which n articles
present the first aspect, n 2 the second aspect, etc., n c the
c th aspect is
P.*'
//it/i/it / i * * * *
n^n^. n c \
the first factor being the number of complexions consistent
with these aspects.
The total number of possible complexions, being the
number of separate terms in a product of n factors each
containing c terms, viz.,
(a n + a 21 + . . . + a cl ) (a 12 + a 22 + . . . + a c2 ) . . .
STATISTICAL METHOD 11
is, of course, c n . On the other hand, the number of statistical
states is the number of terms in the expansion
K + a 2 + ....+ a.)*
which is, therefore, the number of " homogeneous products "
of the c quantities a l9 a 2 , . . . , a c , each product being of the
7i th degree. Reference to a text-book of Algebra will show
that this is
(' + *- 1 )' .... (1.2.6)
V
1 . 3 The Normal Law of Errors. We shall now consider a
modification of this random throwing of a number of articles
which will lead to the introduction of a mathematical
function which at a later stage plays an important part in
statistical-mechanical reasoning.
Instead of coins or dice, let us have in our possession n
counters, each one being labelled + e on one side and
on the other. On throwing these, any complexion will, if
we add the numbers showing, yield a sum me where m is a
positive or negative integer lying between n and + n.
In fact m is r s where r counters show positive faces and
s counters show negative ; thus the probability of the sum
me is given by the number of complexions corresponding to
(r, s). It is
" .... (1.3.1)
* ;
rls
where
r -j- s n
r s = m
By a well known result, the expression ( 1 . 3 . 1 ) is maximum
when r = s, and the value decreases progressively to the
amount (\) n as the difference r s or s r increases in
numerical value from zero to n. Thus the state in which the
sum is zero is the most probable, and if n is a very large
number, the probability of those states in which the sum is
zero or only a small multiple of e, far outweighs the prob-
ability of those in which the sum takes a relatively large
numerical value. It will be both interesting and serviceable
12 STATISTICAL MECHANICS FOR STUDENTS
to investigate the limiting form for this expression (1.3.1)
when is made very small in value and the number n grows
without limit. This condition is, however, rather vague,
for it makes no provision for the maximum value of the
sum, viz., nc. We can make the condition sufficiently
precise by postulating that ne may also increase without
limit as e decreases and n increases, but in such a manner
that the product of the maximum value of the sum, nc,
and the common difference, 2e, between consecutive values
of the sum remains finite and constant. Thus write
where k is a finite constant. So as not to interrupt the
general course of the reasoning we relegate some rather
tedious mathematical steps to an appendix where it is shown
that in the limit the expression (1 . 3 . 1) is equal to
where z is written for (r - s) e and 8z = 2e.
Thus it appears that the probability that the sum lies
between the values i\ and <T 2 , is given by the expression
A exp(-=-,)dz . . . (1.3.3)
It is a well known result that the value of the integral in
(1.3.3) between the limits oo and + oo is kn*, and so the
expression (1.3.3) when Cj is oo and C a is + is unity,
as, of course, it must be since the sum of the probabilities of
all possible states must be unity.
The integral
plays an important part in statistical-mechanical analysis,
as we shall see later, and a number of its most useful pro-
perties are summarised in the appendix. But before passing
* The series
1 + L + 2! + j -f . . . . + -f . . . . ad. inf. v
11 2! 31 n! Jy
is written exp (y). It is, of course, equal to ev, if y is a real quantity.
STATISTICAL METHOD 13
on to the application of the results of this chapter to mole-
cular systems, it may be as well to point out that we have
in this section been treating an important case in the Theory
of Errors of Observations.
We may assume that the accidental error in an obser-
vation made with a definite instrument is the algebraic sum
of a number of component errors of the instrument due to
change in external conditions, uncorrected errors of the
instrument and peculiarities of the observer. Each of these
components may in its turn be regarded as due to a large
number of elementary causes, so that we are not violating
any obvious truth in assuming that an accidental error of
observation is the algebraic sum of a very great number of
very small errors. Suppose we make a further simplifying
assumption (which can only be justified by subsequent
comparison of the results of making it with the facts) that
all these elementary errors have the same numerical magni-
tude, but are as likely to be positive as negative. Thus if
there are n elementary errors altogether, each of magnitude
e, the actual error will in any observation have the magnitude
(r s)e if r of the elementary errors are positive and s are
negative in that observation. But such an outcome will
have the same probability as in the case of the counters,
where r came down positive and s negative. That is, the
probability of the error (r s)e is given, by (1 . 3 . 1), and
proceeding as before we find that if the true value of the
quantity observed is a, the chance that an observation gives
a value lying between a + C i and a + f 2 * s indicated by the
expression (1.3.3). But those interested may refer to the
appendix for further information. It is more to our purpose
to proceed to the study of molecular systems, and the reader
can glance at the appendix when at a later stage he is com-
pelled to know something about the integral.
14 STATISTICAL MECHANICS FOR STUDENTS
APPENDIX TO CHAPTER I
ON THE NORMAL LAWS OF ERRORS
THE problem raised in section (1 . 3) is to obtain a function
of a continuous variable z which shall replace
n\
-(-V
r! \2/
this being a function of a quantity (r s)e which varies by
discrete amounts 2e. Implicitly we have somewhat changed
our point of view. Previously an error which was not an
integral multiple of c, was not supposed to occur at all ; its
probability was zero ; probabilities were, so to speak, con-
centrated on definite errors. Now all errors are possible
from z = oo to z = + oo, and, what may appear para-
doxical at first sight, the probability that the error may
have any definite value is zero. But this is only natural as
the number of possible errors now is unlimited, and the
chance for any one of them is one in infinity, or nothing at
all. The form of the statement must now be that there
exists a function of z,/(z), such that the probability that the
error shall lie between the values z = d and z = C 2 will be
given by the integral
rf.
/(*) d*,
J &
and to find /(z) we must proceed on the assumption that
there is an approximate equality between
/(z)>
and
n\ /l\ n
H7T\2J
where z == (r s)e and 8z = 2e, the approximation being
closer and closer the smaller e, and therefore the larger r and" $
(and, of course, n) for a given value of z. The reader is
warned against a too common misconception that /(z) is
the chance of an error z. He is asked to bear in mind that
STATISTICAL METHOD 15
the function which replaces (n \ / r ! 8 !) . 2~ tt is not/(z), but
f(z) Sz. The infinitesimal is as important a factor of the
function as/(z) itself. The probability which was previously
concentrated on a distinct value of z is now as it were spread
over a small neighbouring range of values.
Writing m for r 8, we have
"*">*,-& o
2e/(me + 2e) ~
^ ' "
and also
r + Us - 1!
Hence
f(m e + 2 6) ^ a
/(m e ) " r + 1
and so
/(me + 2 e) -/(me) _._ r - s + 1 ^_ __ m + 1
/(me + 2c
or
/(m e + 2 e) -/(m c) _.__ m + 1
26 2(71 + l)
On writing z for me, 2 for 2e and k 2 for 2ne 2 , and pro-
ceeding to the limit, we obtain
This is equivalent to
d log /(z) _
Hence
= + constant,
iC
or
The integration constant A is to be determined by the fact
that
16 STATISTICAL MECHANICS FOR STUDENTS
since the integral represents the sum of all the probabilities
and must be unity.
Before proceeding it may be as well to state briefly a
number of results concerning the integral
and kindred integrals. These will prove serviceable in later
chapters. By a simple change of variables, x = za*, we can
write for this integral
a -j| &-** fa.
First of all it can be shown that if the integral is taken
between the limits x = oo and x = + oo, we obtain
(^ *
1 e x dx =TT
J ~ co
Since e~ x * is an even function of x t it is also true
e~ x * dx = -77*.
^ o
The value of
~[ e~ x *dx
is a function of 77, which increases from zero to unity as 77
increases from zero to infinity. Tables of this function for
definite values of 77 are printed in text-books on the Theory
of Errors.* If 77 is equal to 0-1 the expression is about
0-1125 ; if 77 is equal to 0-5, it is 0-5205 ; for 77 equal to 1,
the value is 0-8427 ; and by the time 77 has reached 3,
the expression has attained the value 0-99998 ; so that the
integration from 3 to oo contributes only -00002 of the value
to the function of 77. One can realise the truth of this
in a general way by noting that if x = 0-1, e~ x * =
* See for example, Combinations of Observations, Brunt, A table is
also printed in Jeans' Dynamical Theory of Gases.
STATISTICAL METHOD 17
0-99905, while if x = 3, er* = -00012, and beyond this
e~ x * decreases to very minute values with great rapidity.
It can also be shown that
if the index n is an odd integer, and
/oo
x n e~ x ' dx = *,
if ft is an even integer.
The cases of this which we shall require now and at a later
stage are
co
x e-*' dx = ~
[ \
x* e~ x * dx = ~
I 4
/
x 2 e~ x * dx = -
We can now determine the constant A in the equality
above
A p-02*
( A e ,
f(z)=Aexp (_.
\ /C
A p 02*
A 6 ,
where we write a for I/A 2 . For if
18 STATISTICAL MECHANICS FOR STUDENTS
then
since
Thus the chance that an error may fall between the values
z = x and 2 = C 2 is
This law has been deduced with the aid of rather restrictive
assumptions concerning the nature of the causes giving rise
to errors. Actually, rather broader assumptions can form
a starting point for its deductions, and it is found to be
closely followed in many cases. Of course in some circum-
stances other error laws hold ; e.g., if the restriction that
positive elementary errors are as likely as negative be
removed, we cannot arrive at a function symmetrical in
value with respect to the origin.
The reader will naturally inquire as to the part played by
the constant k (or a) in the law. The way in which it was
introduced (as equal to e(2?i)*), gives no clear indication of
this ; but we can easily arrive at a conclusion concerning it
by inquiring into the average error made in a large series of
observations which conform to the normal law. The equal
preponderance of positive and negative elementary errors
shows that the average of the observations is the true value
of the quantity observed, denoted by a. The chance that
an observation lies between a + z and a -\- z + 8z is
(a/7r)* e~ a * f Sz. Now as negative and positive values of the
error z are equally likely, the average error in the strict
algebraic sense is zero ; but this is of no help to us. However,
we can find the average numerical value of the errors if we
multiply the numerical value of z by its probability (a/7r)*
er** 82, and sum over the whole range, i.e., integrate from
Oto oo.
STATISTICAL METHOD 19
7T/ a
1
Another important manner of estimating an average
value for the error is to square each error, multiply this by
the probability in each case, and sum over all the errors.
This result gives us the mean square of the errors, and the
square root of it is called the " mean square error " (M.S.E.).
Thus we find
-*' dz
= *!
2
and so the M.S.E. is &/2*, and the ratio of the average
numerical error to the M.S.E. is (2/7r)* or 0-798.
Still another result concerns what is called the " median
error," which has such a value that if we denote it by r,
then half the errors lie between r and + r. This quantity
is determined by the equality
02
20 STATISTICAL MECHANICS FOR STUDENTS
or - =0-5.
From the tables of 1 e""*" dx, it can be found that m* has
JG
the value 0-4770, or r = -4770 A;.
It is clear then that the value of the constant k indicates
the standard of precision of the measurements made. The
smaller k is the more accurate has been the series of
observations. Of course all this implies that care and skill
have already been expended on the actual observational
work. No amount of tinkering with the theory of errors is
going to draw reliable conclusions from careless measure-
ments.
CHAPTER II
THE STATISTICS OF A SIMPLE MOLECULAR SYSTEM
2 . 1 The Complexions of a Molecular System. Configura-
tions. We begin our statistical-mechanical investigations
by applying the methods developed in the previous chapter
to a simple body which is conceived to be an aggregate of
molecules, each molecule being regarded as having no
structure, but merely possessing minute mass and extension
in short the ' ( particle " of dynamical theory. Any obvious
changes in the body must, of course, be associated with
unobservable but yet actual changes in the relative con-
figuration and motions of the particles ; but, to be sure, a
great deal of change might be going on in the relative situa-
tions and movements of the molecules without any observable
difference manifesting itself in the general appearance and
behaviour of the body.
The configuration of the molecular system can only be
precisely defined by assigning definite co-ordinates to each
particle (the particles for the moment being regarded as
equivalent to points), and as the configurations possible for
a given volume of the body are therefore unlimited in number
it is impossible to deal with statistical states by counting
the number of configurations consistent with each state,
since these are uncountable. In the Theory of Errors we
regard an error as having a calculable chance if it lies in an
assigned range of errors ; so in the example before us we
regard a complexion of the molecules as defined by assigning
a range of configuration. In short, we do not say that a
particular molecule is at a definite point (x, y, 2), but that
its co-ordinates lie between x and x + 8x, y and y + 8y, z
and z + Sz where Sx, Sy, 8z are assigned small quantities.
In other words, we divide the volume occupied by the body
into a, finite number of " cells," and state that such and such
21
22 STATISTICAL MECHANICS FOR STUDENTS
molecules are at the moment situated in such and such cells.
As far as the counting of complexions is concerned, we
regard the molecules as analogous to the coins or dice of the
previous chapter. The existence of a molecule in an assigned
cell is analogous to the exhibition of a certain aspect by a coin
or die. So long as a molecule remains in the same cell, its
" aspect " is unchanged. A " complexion " of the system
is defined by the particular aspects shown by the individual
molecules, i.e., by the way the individual molecules are
distributed among the cells. Merely to move the molecules
about within the cells does not alter the complexion ; but
the transfer of molecules from cell to cell will alter the com-
plexion even although it is only an interchange which leaves
the number (but not the individuality) of the molecules in
the cells unaltered. The smaller the cells and the greater
their number, the finer is the detail, so to speak, which
distinguishes one complexion from another, but in order to
render counting conceivably possible, the size of each cell
must remain finite and the number of them also finite,
though possibly very large. We shall see later that in
carrying on the mathematical analysis after the counting
has been effected, we may have to resort to integrations
which imply infinitesimal elements of volume unlimited in
number in much the same way as in Chapter I. we passed
from summation over a range of discrete numbers to integra-
tion throughout a range of a continuous variable ; but in
the initial stages of the analysis the assumption of finite
cells is necessary for beginning the calculation at all.
If, therefore, there are n molecules and c cells, we find just
as in section (1.2) that the number of complexions in which
n v molecules are in the first cell, n 2 in the second cell, etc.,
...... , n c in the c th cell is
- _J^ --- . . . (2.1.1)
We shall in future denote this number by the functional
symbol
W (n lt 2> ....... a ).
All this is, of course, a mere matter of counting. Many of
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 23
these complexions would entail the crowding of the body
into one small portion of the volume within its external
surface. Indeed there are c complexions in which all the
molecules would be in one cell. But this consideration need
not deter us from proceeding ; for if we make the proviso
that the cells though small enough to make c a large number,
are yet large enough to contain a large number of molecules
at the average density, so that in fact the ratio of n to c is
also large, then it is possible to show that in all but a
relatively small number of -complexions the molecular
density is uniform or sufficiently near uniformity for the
discrepancy to be undetectable by experimental means. It
is, of course, easy to see that W (n ly n 2 , ...... , n c ) is in-
creased in value if in the denominator two numbers, n r and
n 8 , in any pair of factorials are replaced by two other
integers which have the same sum, but are more nearly
equal to one another.
For
(k + l)\(k l)l
= (k + l) (4+Z- 1) ...... (k + l).k\x
k\
>k\k\
and therefore
_
(k + l)\(k-l)\ k\k\
Thus the state in which the density is uniform embraces a
number of complexions which is greater than for any other
state in which the numbers n l9 n 2 , ...... ,n e are given, but
are not all equal to n/c.* But to prove the statement con-
cerning the preponderance in number of the complexions
corresponding to uniform density or states near it over all
other complexions requires closer analysis than this ; and
to proceed with the proof we require the assistance of an
approximation to W (n lt n 2 , ...... , n c ), which, however,
* We suppose the number of molecules and number of cells to be so
chosen that n/c is an integer.
24 STATISTICAL MECHANICS FOR STUDENTS
has to be used with due consideration for the conditions
under which it is true.
There is a famous theorem published early in the eighteenth
century by the Scottish mathematician James Stirling,
which states that there is an approximate equality between
the logarithm (to the Napierian base) of the factorial of n
and the expression '
(n + 2 j log, n - n + ~ log, 2 IT.
Provided n is as large as 10, for example, a four-figure
logarithm table will not reveal the discrepancy, and for
sufficiently large numbers, it is possible to write
log n \ = n log n - n.
(See the appendix for some remarks on Stirling's theorem.)
From this it follows that, provided none of the numbers
n r are too small,
c
log W (n l9 n 2> , n c ) = n log n 2 n r log n r (2.1.2)
r=i
With the aid of this approximation one can prove the
assertion made above. The details of the proof will be
found in the appendix to this chapter, and the outcome is to
demonstrate as we have already said, that the statistical
states in which the molecules are uniformly distributed, or
distributed in such a manner that the number in each cell
differs but little from the average, embrace all but a negli-
gible fraction of the total complexions. The point of this
important result will appear presently.
2 . 2 The Complexions of a Molecular System. Phases.
In the previous section we confined our attention to the
arrangements of the molecules in space ; but, of course, in
any attempt to explain the general properties of matter by
dynamical theory, we must also take account of the motions
of the molecules relative to the frame of reference in which
the body as a whole is regarded as fixed. Since the molecules
are regarded as particles, it is easy to conceive a graphical
representation of a precise velocity-condition. Choosing an
origin, the velocity of a particle can be represented by the
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 25
position of a point in a " velocity-diagram, " the vector from
the origin to the point representing the magnitude and
direction of the velocity. Thus the actual velocities of the
n particles at any time are represented by a configuration in
the velocity -diagram of the n representative points. This
diagram can, of course, be divided into cells as in the case
of the volume occupied by the body ; velocity-complexions
can be defined just as before and the number of complexions
embraced in any velocity-state (where velocities within
narrow limits are assigned without reference to the indi-
viduality of the molecules) can be counted and comparisons
of relative numerical strength be made.
It is customary to link up the two methods of partitioning
into one " picture." The configuration and velocity of each
molecule is' regarded as an entity with six components, three
position-components and three velocity-components, and is
called a " phase " of the molecule. When precise values of
velocity and position are assigned to each molecule in the
system, we are said to have prescribed a " phase of the
system/' As before, the phases possible to the system are
unlimited in number, and no progress can be made in
counting complexions unless a complexion is regarded not
as a phase, but an arrangement of the molecules in a small
but finite " extension-in-phase." That is, we do not desig-
nate an " aspect " of a molecule by saying that its position
and velocity are given by components x, y, z, u, v, w, but by
saying that the first co-ordinate lies between x and x + Sx,
etc. ; the first velocity-component between u and u + 8u
etc., where Sx, , Sw are finite but small increments.
We can visualise as small rectangular figures the forms to
which we attach the symbols, and regard 8x By Sz and
Su 8v Sw as their volumes. With no possibility of visualising,
we nevertheless refer to the magnitude Sx Sy Sz Su Sv Sw as
an element of " extension-in-phase," and for convenience
and brevity call it a " phase-cell," borrowing the geometric
term from our earlier considerations. Indeed, in a great
deal of the literature of the subject, geometrical language is
used in a manner which at the outset may dismay the
beginner, who imagines he is called upon to perform the
26 STATISTICAL MECHANICS FOR STUDENTS
feat of " seeing " a space of six or more dimensions. He
may reassure himself ; no such impossible task is expected
from him. The geometric terms are merely borrowed and
attached to analogous ideas ; indeed their use can be avoided
altogether (e.g., that is the practice of Gibbs), but for the
sake of reading the general literature later, the beginner
should familiarise himself with this use of geometric terms.
For instance, a phase of a molecule which is indicated by
assigning definite values to the x, y, z, u, v, w, of the molecule,
is referred to as a " point in the six-dimensional phase-
diagram/' An extension in phase, which is a range of values
of position and velocity co-ordinates such that no phase in
the range has co-ordinate values outside the six ranges x to
x -\- 8x, w to w + Sw, is called a cell of the phase-
diagram. We can conceivably visualise the path of a mole-
cule in physical space. It is the geometric counterpart of a
continuous series of values of the position co-ordinates. We
can also visualise a curve in a velocity-diagram which would
represent geometrically the changing values of u, v, w, for a
molecule. Indeed the reader has perhaps met it in his
academic text-books of mechanics under the name " hodo-
graph." When we place the two pictures together, we
cannot get a " picture," but we still use the geometric
language ; the series of continuous phases through which
the position and motion of a molecule pass with lapse of
time, is called the " phase-path " or " trajectory " of the
molecule.
If the reader still experiences any difficulty in this matter
of geometric terminology, he may find the following device
of some assistance, until he becomes sufficiently familiar to
dispense with it. Let him think of three plane diagrams,
each one provided with the usual pair of rectangular axes.
In one let x and u be represented by a point ; in the second
let y and v be so represented ; and in the third, z and w. A
phase of a molecule can then be visualised by thinking of
three points one in each diagram ; an extension in phase
by thinking of three small rectangles ; the changing phases
of a molecule by thinking of three curves. A " point-group "
represents a phase ; a " curve-group " represents the history
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 27
of the molecule and a " rectangle-group " represents an
extension-in-phase. The division of the phase-diagram into
cells may be associated in the mind's eye with the cross-
meshing of the plane diagrams by lines parallel to the axes.
One has to be careful, however. If even one member of a
point-group is changed, the phase is altered ; we have
changed to a different point in the phase-diagram. Keeping
two rectangles in the group alike, but changing the third,
produces a different phase-cell. Suppose, for example, we
have divided a limited portion of the (x, u) diagram into I
rectangles, another limited portion of the (y, v) diagram
into m rectangles, and of the (z, w) into n rectangles, we are
dealing with a limited extension in the phase-diagram which
has been divided into I m n cells ; for any rectangle in the
(#, u) diagram can be combined with any in the (y, v), and
this again with any in the (2, w), each combination pro-
ducing a distinct phase-cell.
The algebraic work from this point is just as before. If
there are c phase-cells and n molecules, the number of com-
plexions embraced in that state in which the phases of n l
molecules are within the first cell, etc., of n c molecules
within the c th cell is W (n l ,n 2 , n e )
where
W (i' > ' ra c) = ,_, rr
'H- n 2' U C-
and approximately
c
log W (n l9 rc 2 , n c ) = n log n 2 ri r log n r .
r=l
So far there has been no reference to dynamical con-
siderations. We have been concentrating our attention on
the concepts necessary for the statistical side of the work.
We must now turn our attention to the second member of
the double barrelled epithet, " statistical-mechanical/'
28 STATISTICAL MECHANICS FOR STUDENTS
APPENDIX TO CHAPTER II
A. Stirling's Theorem. A rigorous proof of this theorem,
which states that
can only be appreciated by those who have given some pains
to the study of series and their convergency properties.
Such a proof will be found by those interested in ChrystaFs
Algebra, Vol. II. Chap. XXX. For a more general theorem
which takes the place of Stirling's when n is not an integer,
a reference can be made to Whittaker and Watson's Modern
Analysis, Chap. XII.
For the majority of his readers, the author suspects that
a simple empirical test will be quite satisfactory. If any
one cares to take the trouble to look up Napierian logarithms
in a book of tables (the cheap little book prepared by C. G.
Knott will suffice), he will find that for instance
log 5 - ^g 1 + log 2 + ...... + log 5
* 5
has the value 0*6519. He will also find that
log 10 + log 77
1 -
10
has the value 0*6553. If he chooses a larger number 10, he
will find that
l og 10 - lQ g * + lQ g 2 + + lQ g 10
^ 10
has the value 0-7922 ; and that
- 1 __ log 20 + log 77
20
is also 0*7922, so that a four-figure table cannot distinguish
between the values of the two expressions. This empirical
procedure gives considerable support to the general result
that
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 29
* + lo 2 + ...... + l n
log n
ft
_., log 2 ft 4- log
2ft
or
log ft ! == ft log ft ft 4~ - log (2 77 ft).
J
It follows that
.e
A rather more general (but still far from rigorous) pro-
cedure is to begin from the well-known integration theorem
I log x dx = x log x x -\- constant.
Taking this between the limits 1 to ft, we have
fn
I log x dx = ft log ft ft 4~ !
We can replace the integral by the approximate expression
(x 2 x ) log x + fc 3 # 2 ) log# 2 + . . . 4- (x f x f __i) log x f _ l9
where x v x 2 , x s , , x f are a series of equally spaced
values of x ranging from 1 to ft , provided any of the differences
x r x r _ l are small compared to the x r . If ft is very large,
we can makes these differences unity, and thus replace the
integral by
log 1 4- log 2 4- + log ft.
Thus approximately
n
E log r === ft log ft ft -f 1
i
= ft log ft ft.
B. The Preponderance of Certain States as Regards the
Numbers of Complexions Embraced in Them. We have seen
that in a state in which the distribution of the molecules
among the cells is n v n 2 , ...... , n c , the " complexion-
number " W (ft 1? ft 2 , ...... , ft c ) is given by the approxi-
mation
e
log W (n l9 ft a , ...... n c ) = ft log ft 2 n r log ft, .
r-l
30 STATISTICAL MECHANICS FOR STUDENTS
We shall denote the expression on the right-hand side by
k (n v n 2 , ...... , n c ). The function k (n lt n 2 , ...... , n c )
has its maximum value when all the n r are each equal to
n/c \ this value we shall denote by k m . So the maximum
value of W (n v n 2 , ...... n c ) is W m where
lo g w = k m = n (log n log a),
a being written for n/c.
To investigate the relative numerical strength of states
in the neighbourhood of this state, let us write
n 1 =a+p l ,n 2 =a+ P 2 , ...... , n c = a + &,
where p l9 /? 2 , ...... , p c are a set of positive or negative
integers which must satisfy the relation
r-1
It follows that k (n v n 2 , ...... n c ) for this state is con-
nected with k m by the equation
k m -k=Z\(a+p r )log(a+p r )\ -nloga
using the well-known expansion of log (1 + x).
If we work out this expression we obtain a series whose
terms are multiples of the expressions 2 /? f , 27 $, 2 , S r 3 ,
etc. The first term is, of course, zero, and we find after a
little rearrangement that
-Tn^l) +etc '
10r=i\ay
For small values of the /? f , the series reduces to its first term,
which, being a sum of squares, is essentially positive, as we
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 31
should naturally expect since k m > k for any values of the
&
In the text of the chapter, the statement was made that
the number of complexions embraced in these states for
which the jS r are zero or equal to small fractions of a, are
sufficiently great to " swamp " the complexions embraced
in states diverging in a perceptible degree from uniformity
of distribution. The justification of this depends on equation
(I.). On its right-hand side the ratio of the second term to
the first is
1
3 a
and if the ratios of the various fi r to a are sufficiently small,
this is also small, so that the second and also the subsequent
terms of the series can be neglected, and we can write
___ ^ __ y r 2
K -^ ^ x r *
where
We thus arrive at the result that if W is the complexion
number for the state defined by the integral values f3 l9 j8 2 ,
j8 c for deviation from uniformity, then W satisfies
approximately the equality.
W = e*
aR*\
J
where
W, of course, only attaining the maximum value W w where
every j3 r (or x r ) is zero.
Now let us for the moment turn our attention to the follow-
ing little problem. Despite its apparent irrelevance, it will
soon be apparent why we do so.
32 STATISTICAL MECHANICS FOR STUDENTS
Imagine that we have a series of small particles dotted
about in space at the points of a regular cubical lattice, and
that the mass of any particle is equal to m e~ ar * where r
is the distance of the particle from an origin which is itself
a point of the lattice, m being the mass of the particle at the
origin. If we were required to find the total mass within a
sphere of given radius Z, and the elementary cubes of the
lattice were small enough, a sufficiently accurate answer to
the question would be found by supposing the matter to be
distributed continuously and not at discrete points, so that
the density at a point is given by p e~ ar \ p being the
density at the origin. The mass within a sphere I would be
r 2 e~ ar dr,
J
which is equal to
f 1
r
Jo
where
= m*
Now we saw in the appendix to the last chapter that the
definite integral
is equal to ?r*/4 ; it is also true that the whole of this value
is practically contributed between the limits to 3 ; which
shows that in our problem almost the entire mass of the
lattice of particles is within a sphere whose radius is three
times a""*.
To return, after this digression, to the main argument,
the solution of the problem concerning the preponderance
of complexions is just a multidimensional analogue of the
lattice problem with an analogous answer. A state can be
supposed to be represented by a " point " x v x%, , x c
in a c-dimensional space. At the " point " we suppose
some entity of magnitude W located. We can then, in order
to reduce the mathematical process to an integration,
suppose the entity distributed uniformly with a density
STATISTICS OF A SIMPLE MOLECULAR SYSTEM 33
A C
A exp [ -
P \ 2
and the amount of the entity within a certain extension of
the c-dimensional space can be determined by the integral
A I ...... \ e~ ~*~ dx l dx 2 ...... dx c
throughout the extension.
The knowledge of integration required is rather outside
the usual academic courses delivered to students for whom
this book is intended. Those interested will find the neces-
sary material under the heading " Dirichlet's Integrals " in
the more advanced texts on mathematical analysis, and the
full working out of this problem in Jeans' Dynamical Theory
of Gases. The upshot is similar to that of our lattice problem.
Practically the whole W-magnitude of the entity is within
an extension close round the origin, that is where none of
the x r exceed rather small integral multiples of (2/a)*.
Since a is assumed to be a large number, this implies that
practically all the complexions possible to the molecular
system are concentrated in states for which the j8 r are small
compared with a, i.e., states in which deviation from
uniformity is not marked.
It is true that in the above the approximations which we
have made exclude implicitly from the analysis states where
there are only a few molecules in some cells, since the par-
ticular use of Stirling's theorem is rather wide of the mark
for small values of n r . However, a simple consideration of
the original factorial expressions will show how relatively
unimportant such states are ; and in any case one can use
the still closer approximation for log n\, viz., (n + ^)
log n n + ^ log 2 TT, and find the conclusion still justified,
although the expressions are a little more complicated.
CHAPTER III
THE PROBABILITIES OF THE DIFFERENT STATES OF A
SIMPLE MOLECULAR SYSTEM
3 . 1 The a priori Probability of a Complexion. So far we
have been confining our attention to the counting of com-
plexions. In order to discover the probability of a state,
we clearly must know or postulate something about the
probabilities of the complexions included in it. But, it must
be admitted that here we are faced with one of the most
difficult questions in the whole subject. There is nothing
novel, however, in such a situation. A satisfactory settle-
ment of the postulates of any of the mathematical sciences
fih^i proved to be one of the most difficult tasks, calling for
intellectual endowment of no mean order on the part of
those who have been the pioneers in this elusive branch of
knowledge. Many a young student acquires a really skilful
mathematical technique during his school and university
years, without being aware of the doubtful nature of some
of the processes employed unless hedged round with an
array of carefully-phrased conditions. Not that there is in
this state of affairs anything calling for severe censure on
the part of the educational reformer. It is doubtful whether
much good can come from too much immersion in " founda-
tions " on the part of a mind as yet immature. No sane
teacher ever dreams of troubling boys and girls with closely-
reasoned disquisitions on the postulates of geometry.
Certain statements very plausible to the young concerning
congruence, parallelism and the like, are accepted at the
outset, and a start is made on the deduction of theorems,
arousing immediate and practical interest in the minds of
the pupils.
The author feels that the situation at this stage of our
subject is essentially similar. He surmises that his readers
34
PROBABILITIES OF STATES 35
are more anxious to " see results " as soon as possible,
without being worried at the moment with a logical dis-
cussion of postulates, which, if they have any element of
plausibility about them, will be accepted without demur,
especially if they lead to conclusions in agreement with
experimental fact. Such a posteriori justification may
indeed be entirely satisfactory for many ; but of course it
would not be sound to leave matters in such a condition,
and some attempt will be made at a later stage to discuss
the basis of the postulates used. This, however, cannot be
done until the mathematical expression of dynamical laws
has been carried a stage further than is usual in the ele-
mentary texts of dynamics.
The fundamental postulate introduced at this stage is
that if the various phase-cells have the same magnitude,
one aspect of a given molecule is as probable as any other ;
i.e., it is as likely for its " representative point " in the
" phase-diagram " to be in any phase-cell as in any other.
There is a certain plausibility about the assumption, which,
however, must not blind us to the fact that some time or
other it should be subjected to careful scrutiny. Clearly the
history of the system is determined by dynamical law, and
if our postulate be true, it must at least be a possibility that
any molecule can in a sufficiently long lapse of time have
been in every cell. In the first chapter the postulate of
equal a priori probability of each aspect for an article after
a throw was based on the fact that repeated throwing does
show every aspect for an article in approximately equal
numbers during a long time. No such direct experimental
evidence comes to our assistance here ; we are not on such
familiar terms with individual molecules as with coins and
dice ; nor have we the services of " Maxwell demons " to
call upon ! However, let us make the assumption for the
sake of practical progress now and worry about it later.
Certain obvious reservations must be made in the appli-
cation of this postulate, however, which call for immediate
notice. If the body of which our molecular system is regarded
as an analogue, be solid, we find it hard to admit that any
molecule can be in any " configuration-cell " into which we
D 2
36 STATISTICAL MECHANICS FOR STUDENTS
divide the volume, even in long lapses of time. No doubt
\ve can appeal to the experiments of Roberts -Austen and
others on diffusion in the solid state ; but still the essence
of the molecular picture of a solid is that the molecules,
even if they are not fixed in unchanging neighbouring
positions relative to one another, are vibrating about the
points of some lattice as origins. And if each molecule
cannot wander from configuration-cell to configuration-cell
at random, neither can its representative point in the phase-
diagram roam fortuitously about the phase -cells. This
difficulty, however, is not serious. It serves to show that
we shall have to treat the solid state on a somewhat different
plan to the fluid states where our postulate has a greater air
of plausibility. We shall find that in the solid state the
interest centres round the vibrations about the set of fixed
points and the co-ordinates and velocities of each molecule
with reference to its own individual " origin " are the
quantities which determine the phase at any definite instant,
and it is with regard to partitions of a phase-diagram con-
structed on such an understanding that our postulate will
be introduced.
For the present, therefore, we confine ourselves to the
fluid states, and in order to simplify our preliminary con-
siderations we shall begin with the gaseous state where no
difficulties will be raised by intermolecular forces. In short,
we shall at the outset deal with a monatomic gas.
Another condition which restricts our postulate concerns
the question of energy. In partitioning into configuration-
cells, a natural limit is placed on the cells of a configuration-
diagram by the external surface of the body. In a velocity-
diagram no such boundary is obvious, yet dynamical con-
siderations yield one very readily. If a velocity-cell were
sufficiently far from the origin, the velocity represented by
its central point" might be so great that if even a single
molecule were possessed of this velocity, its kinetic energy
would be greater than the energy-content of the body con-
sidered. Thus a sphere of definite radius excludes velocity-
cells which cannot come within our consideration. Indeed,
instead of appealing to a hard, impenetrable boundary to
PROBABILITIES OF STATES 37
limit the configuration-cells, we might adopt a similar
method of limitation as for the velocity -cells. That is,
conceive that the region occupied by the gas is a region in
which a field of force exists whose potential is vanishingly
small unless one approaches the boundary, where it rises
rapidly as we proceed outwards to values so great, that were
even one molecule to be in this external region, its potential
energy would exceed the energy-content of the body. Thus
we place a natural limit on the phase-cells, to which the
postulate of equal a priori probability can be applied, by
means of one condition, viz., the definite energy-content of
the body of which the molecular system is a model.
One other point should be mentioned, before proceeding
to definite probability calculations. For a reason which
may not appear necessary at present, but which will be
recognised as a matter of great convenience later, we will
assume that the components of the phase of a molecule will
include not the components of its velocity, but of its
momentum. Of course, at present, this amounts to nothing
more than a change of scale in the phase diagram ; but the
change will justify itself very decidedly when dealing with
more complex systems.
If there are c cells of equal magnitude in the phase-
diagram, the probability of a molecular representative point
being in one of them, is by the postulate of equal probability
equal to c" 1 ; i.e., in the formula (1 . 2 . 4)^9 1 ~p%
= p c = c" 1 , and the probability of the state (n ly n 2 ,
T& C ), would appear to be
W (n lt n 2 , ,n c ) c~ n .
But this overlooks the limitation placed upon the total
number of possible complexions by energy considerations.
If the body of which the molecular system is a model is
considered to be at constant temperature, the energy-content
is given, and the total number of complexions is not c w , for
many of these would be inconsistent with the equations
n l + n 2 + + n c = n . . . (3.1.1)
n l l + n 2 2 + +tt c c = E. . . (3.1.2)
38 STATISTICAL MECHANICS FOR STUDENTS
where 19 e 2 , , e c are the energies of a molecule when
its representative point is in the first, second, , c th
cell respectively and E is the energy-content of the body.
We have to consider the sets of positive integers n l9 n 2 ,
,n c which satisfy (3. 1 . l)and(3. 1 .2), arid add together
the W (n lt n 2 , n c ) functions for each set. If the sum
is s, then s is the total number of possible complexions, and
the probability of the state (n l9 n 2) , n c ) is
W (n v ?2 2 , . . .,n c )
s ... (3.1.3)
These considerations clearly take account of the condition
mentioned above which determines the boundary of the
possible phase cells ; for any cells for which e > E are
excluded automatically by (3.1.2). The quantities e r are
the energies corresponding to the centres of the cells. They
will be given by some expression such as
e -i- ^ + c 2 , , , ,
- 2 +^ (w) '
where , ??, Care the components of momentum for the phase,
m the mass of the molecule, and $ (x, y, z) the potential
energy of the molecule in the phase, due to any external
field of force, such as gravity. For certain phase-cells on the
boundary <f> (x, y, z) will also include the potential energy,
mentioned earlier, which has an evanescent value in the
main portion of the body's volume, but becomes rapidly
significant as we approach the external surface, rising to a
value greater than E. In the functional form of </) (x, y, z)
will occur certain coefficients entering as factors of the
powers and products of x, y, z in the various terms or in
some other recognised manner. E.g., the quantity g will
occur if gravity is regarded as acting on the molecules.
These quantities are called " parameters/' They are
regarded as constants in investigations on the probabilities
of given states for a system with given energy, but, as will
appear later, when drawing thermodynamic conclusions
from statistical mechanics we have to consider changes in
energy -content, and one way of producing such change is
by an alteration of parameters, such as is caused by a change
PROBABILITIES OF STATES 39
in external bodies producing a field of force or in a move-
ment of the external surface of the body ; this, in fact, being
the analogue of " external work " done by or on the system.
The reader is reminded that we are at the moment dealing
with a gaseous system where intermolecular forces play no
part. If it were not so, the energy of the system would
include the mutual potential energy of the molecules which
would not be proportional to the first power of the con-
centration and in the expression on the left side of (3.1.2)
the r could not be regarded as merely functions of the phases
and parameters, but would also depend on the values of
n l9 n 2 , , n c . In other words, E could not be put
equal to a linear function of the n r . This consideration must
seriously modify the treatment of liquids as compared with
that of gases.
The expression (3.1. 3) clearly implies an equal probability
for each of the s possible complexions, and we refer once
more to the necessity for a closer scrutiny at a later stage
of the postulated foundations. This is no matter of repeated
random " throwing " of molecules into cells. In casting
dice or coins they are gathered up and thrown in a manner
subject to no law other than the " law of chance." But
even if we imagine that the molecular system is just now in
any complexion we like, the subsequent complexions of its
history do not follow " by chance," but are the results of
movements and collisions subject to the law r s of dynamics.
Waiving, however, closer investigation of this knotty point
for the present, we proceed to discover the state with the
maximum probability.
3 . 2 The State of Maximum Probability. This will no
longer be given by the equality of n v n 2 , , n c . The
condition (3 . 1 .2) alters the whole character of the solution.
As before, we have to find the values of the n r which will
make W (n ly n 2 , , n c ) maximum (for given E, s is
also given), but subject to the condition (3.1.2) as well as
(3.1.1). Postponing for the moment the details of the
solution, it appears that in the most probable state n l = v^
n% = v 2 , , etc., where
v r =Ce~^ . . . . (3.2.1)
40 STATISTICAL MECHANICS FOR STUDENTS
C and JJL being two quantities which are functions of E, n
and the parameters. These are determined by the two equa-
tions (which are, of course, conditions (3.1.1) and (3.1.2)
for this state),
c
2 e-^r = n . . . (3.2.2)
c
C S r e-^'r =E . . . (3.2.3)
r 1
Division of (3 . 2 . 3) by (3 . 2 . 2) gives
Z e r er^r E
2 e' 1 "' n
(3.2. 4).
Equation (3.2.4) determines /x as a function of the average
molecular energy E//& and the parameters, and thereupon
(3 . 2 . 2) or (3 . 2 . 3) will determine C.
When we set out the steps of the solution presently we
shall see that, since it follows the usual procedure of deter-
mining maxima and minima in the calculus, we are treating
the n r as continuous variables for the time being and, of
course, there is no guarantee that any of the expressions
C* e~"^r are integral. This is one of those minor troubles
which beset us when we are engaged in discussing bodies
with molecular structure by a mathematical method which
practically implies that we are dealing with a continuous
medium. The consequences, as we shall see, are in certain
connections too serious to be overlooked and will compel us
to adopt some form of " quantum hypothesis/' but in the
present connection we are after all dealing in the main with
such large values of n r that a change of unity while not
exactly infinitesimal is relatively so small that the results
of introducing the infinitesimal variations of the calculus is
not going to lead to serious error. In (3 . 2 . 1) we can regard
v r as such an integer that C e~* f r lies between and v r and
v r 1, if not exactly equal to v r . The form of the solution
once more excludes certain cells without any appeal to
physical boundaries ; for if e r is large enough, e~^ e f is so
small that C e~^ f f is a proper fraction. It follows that in
(3.2.2) and (3.2.3) the limits of the summations may be
PROBABILITIES OF STATES 41
omitted and the series regarded as infinite since beyond a
certain term there will be a negligible residue.
A proof exactly on the lines of that in the appendix to
Chapter II. can be constructed to show that if we consider
the probabilities of states given by
n r = r + &
then the combined probabilities of the most probable state
and those states for which the /3 r have relatively small
values practically swamp the probabilities of all other states.
Thus, subject to a satisfactory settlement of the doubtful postulate
of equal probability of individual complexions in the prolonged
history of the gas, we find that there is a " normal state " of
the molecular model in or near which it will always be,
except for brief and insignificant intervals of time. It is
this state which we clearly must investigate if we are to
derive by statistical-mechanical methods the well known
thermodynamic properties of a system in thermodynamic
equilibrium. We shall begin this task in the next chapter,
and conclude this one by laying out the proof of the result
(3.2.1).
As before we seek the " max-min " condition for log W
regarded as given by the approximation
c
n log n Z n r log n r . . . (3.2.5)
r = l
subject to the conditions (3.1.1) and (3.1. 2). As already
stated we can without serious error regard the n r as con-
tinuous variables. The method employed is known as the
" Method of Undetermined Multipliers," and although not
absolutely necessary, a perusal of an exposition of the
method in a text of the calculus will prove helpful to any
reader not familiar with it.
Let us alter the n r to n r -f- Sn r , then the variation in
log W is given by
8 log W = - S (1 + log n r ) Sn r . . (3.2.6)
and the variations, Sn r , must, on account of (3 . 1 . 1) and
(3.1.2) satisfy the two equations
ZSn r = . . . . (3.2.7)
Z r 8n r = . . . . (3.2.8).
42 STATISTICAL MECHANICS FOR STUDENTS
But if the values of n r be such as to make log W maximum
or minimum for any small variations from these values, then
the equation
27 (1 + log n r ) Sn, = 0,
or, by reason of (3 . 2 . 7),
Z log n r Sn r = . . . . (3.2.9)
must be true as well as the equations (3.2.7) and (3.2.8).
If A and /z are any quantities whatever, it follows as a matter
of course that
Z (log n r + A + /*,) 8n r = . . (3.2.10)
This being so, let us choose X and /z so that
log n + A + fi l =
log n 2 + A + p, 2 =
This is quite possible ; these are two simple simultaneous
equations determining the two quantities A and /z uniquely
as functions of l9 e 2 , n l and n 2 ; in fact
_ lo^gj- log n^
^ ~ i *z
A =-*i. Iog -^ 2 Jr_ .2 .l?gi
1 2
It follows that with these two values of A and \i
(log n z + A + ^e 3 ) 8^3 + (log n + A
etc. =-0 . (3.2.11)
This result must be true for any values of the variations
8n 3 , 8n 4 , ...... ; for having chosen any set of them, we can
choose 8n l and 8n 2 to satisfy
=0.
But this cannot T3e so unless the individual multipliers of
Sft 3 , 87i 4 , etc., in (3 . 2 . 11) are themselves zero, provided, of
course, that A and //, have the values given above. Thus it
transpires that in the condition when log W, and therefore
W/#, has a maximum or minimum value, the following
equations are all true
PROBABILITIES OF STATES 43
. (3.2.12)
log n
i+A+/x
l =Q
log?i
2 + A + p
e 2 = .
log n
3 + A + p
r =
log n r + A + /i r =
These equations determine the n f as functions of the c r
and A and /z, and if these values are inserted in (3 . 1 . 1) and
(3 . 1 .2) they determine A and JJL as functions of the c r , E and n ;
and so determine the n r as functions of the e r , E and n. In
short, c equations, such as (3 . 2 . 12) and the equations (3.1.1)
and (3.1. 2), constitute c + 2 simultaneous equations deter-
mining uniquely the c + 2 quantities n l9 n 29 n c , A, p,
in terms of the e r , E and w. The result is
n r C e~^v,
where C e~\ and as pointed out above, (3.2.4) will then
determine p, and (3 . 2 . 2) or (3 . 2 . 3) will determine C or A.
Since the equations determining C (or A) and p, involve
summations over all the cells, which in practice will amount
to integrations throughout the region of the phase -diagram
bounded by the energy-condition, C and p, will not depend
on individual e f , i.e., on individual phases, but will be
functions of the parameters which, in conjunction with the
phase, enter into the functional form giving any e f in terms
of , 77,, f r , x r9 y r , z r .
We still have to satisfy ourselves that the corresponding
value of W is a maximum and not a minimum. This is
almost obvious " on sight," but if any one requires a formal
proof, he has only to make a small modification of the
analysis in Appendix B to Chapter II. Writing v r + /?, for
v r , we find
log W m - log W - Z{(v r + j8 r ) log (v r + p r ) - v r log v r ]
(and after a few steps on precisely similar lines to those in
the appendix)
2
]}
44 STATISTICAL MECHANICS FOR STUDENTS
For small values of the /3 r the right-hand side is positive,
no matter what the sign of the /3 r (for, of course, the v r are
positive).
Hence log W TO > log W, and W w is therefore a maximum
and not a minimum. A continuation of the discussion leads
as before to the enormous preponderance of the combined
probabilities of the normal state and states near it over the
combined probabilities of the remaining states.
CHAPTER IV
TEMPERATURE AND THE DISTRIBUTION CONSTANT
4 . 1 The Average Energy of a Molecule. When we begin
to consider some practicable method of calculating p and C
from equations (3.2.2) and (3 . 2 . 3), it is clear that no pro-
gress is possible by methods of series summation unless
some definite information is available concerning the magni-
tude of the phase-cell. There is nothing in the so-called
" classical " dynamical methods to give us any help in this
respect ; in fact it is one of the signal benefits which the
quantum hypothesis has conferred on our mathematical
methods that it has in conjunction with experimental work
on spectroscopic phenomena suggested an answer to this
difficulty. However, we shall have to defer this matter to
a later stage. In the meantime we shall have to convert the
series summations into integrations, thus leaving in our
expressions an entirely undetermined quantity, which, for
many purposes, is no drawback. Thus we shall have to
replace a symbol such as n r by an integral
J ..... J
?> C,x,y,z)dd7]d(dxdydz y
the integration being extended over the r th phase-cell,
/(> ??> C ^) y> z) being some continuous function of the
variables , 77, C, #, y, z. The solution for the state of
maximum probability worked out in the last chapter shows
that for v r we must write the above integral with the function
/ (, ^, C x, y, z) given the form
- (p + ^ + (*)-[*<(> (x 9 y, z)
zm
where D is a constant.
Confining ourselves for the moment to the case in which
45
46 STATISTICAL MECHANICS FOR STUDENTS
the gas is free from external force, it follows that v r is replaced
by
D[ ..... (exp { -a ( z + 7j 2 + P)}d{dT) dtdxdydz (4.1. 1)*
extended over the r th phase-cell, where a = /*/2m. Further
the quantity e r v r must be replaced by
+ ^ + C 2 ) exp { - a\P + ^ + C 2 ) !
d ..... dz (4.1.2)
also extended over the r th cell.
The equations (3.2.2) and (3.2.3) are now replaced by
~~ a + j = n
" a
(4.1.4)
where v is the volume of the system. The summations
were over all phase-cells, so that the integrations are now
practically from oo to co for each variable. Using the
results quoted in the appendix to Chapter I. we obtain
from (4.1.3)
. Dt; (~\ = n . . . . (4.1.5).
To deal with (4,1.4) we first observe that
dC
-i (5)' ever
* Of course t will involve x, y, 2, in the boundary cells where the
limiting field of force acts ; but since is very large in these, no practical
contribution to the integral is made by such cells.
TEMPERATURE AND DISTRIBUTION CONSTANT 47
and so (4.1.4) simplifies to
?^(*Y = E .... (4.1.6)
2m 2 \a 5 / V '
Hence, dividing (4 . 1 . 6) by (4 . 1 . 5), we obtain
3 _ E
4ma n
3n . . . . (4.1.7)
""as
and, in consequence
D =
v
n (4 1 8)
' ' I*' 1 ' 5 '
The result (4.1.7) shows that the ''distribution-constant "
fj, of the system is inversely proportional to the average
energy of a molecule, so that if the temperature of the
system rises, the distribution of the molecules among the
phase-cells in the normal state is altered since //, decreases.
Thus there is a direct relation between p and temperature,
and it can be readily obtained by considering the pressure
of the gas.
4 . 2 The Equation of State of a Perfect Gas. The con-
nection between the pressure of the gas and the velocities of
its molecules is a simple one which requires for its derivation
no other assumption than the postulate that the directions
of the molecular velocities shall be distributed uniformly
between all possible directions. A molecule which crosses
an element of surface within the gas with a velocity u in a
direction making an angle <f> with the surface, transfers
across the surface an amount m u cos < of momentum normal
to the surface. Let N (u, <f>) stand for the number of mole-
cules per unit volume which have velocities within infinitesi-
mal limits of u, and directions of motion within infinitesimal
limits of the direction <f>. Of such molecules those that cross
the element of surface in unit time lie in a volume
A u cos <f>
48 STATISTICAL MECHANICS FOR STUDENTS
where A is the element of area. Such molecules therefore
transfer normal momentum at the rate
A N (w, 0) m u 2 cos 2 <f>
Summing this for all values of (/>, and remembering that
the average value of cos 2 ^ is 1/3,* we find that this rate of
transfer of normal momentum across A due to molecules
with a velocity u or infinitesimally near it, is
- A N (u) m u 2
where N (u) is the number of molecules per unit volume
having velocities u or near it.
This is
where e(^) is the kinetic energy \ m u 2 of a molecule. Sum-
ming for all molecules, we find that the rate of transference
of normal momentum across the surface A is
2 E
\ .
3 v
Thus the pressure is
2 E
3 ~v
or two-thirds of the energy-density.! Hence if p is the
pressure
2
pv =_ E
1 3
by(4.1.7) =* ..... (4.2.1).
P
But the well-known equation of state is
pv = R0 ..... (4.2.2)
* If a, /?, 7 are the three direction-cosines of any direction with respect
to three rectangular axes,
cos 2 a 4- cos 2 j3 -f cos 2 7 = 1,
and so _ . _ _ ^
cos 2 a = cos 2 =B cos 2 7 =-=.
o
t Further remarks on this method of calculating the pressure will be
made in section (4.5).
TEMPERATURE AND DISTRIBUTION CONSTANT 49
where 6 is the absolute temperature and R the gas-constant
for the given quantity of gas. Thus a comparison of (4 . 2 . 1)
and (4.2.2) yields
and since R is proportional to n (for at the same temperature
and pressure the number of molecules is proportional to the
volume), we have
p." 1 =k9 ...... (4.2.3)
where
7 R
k =
n
and is called the " gas-constant per molecule."
The simple connection between the distribution-constant
and the temperature of the gas is now apparent. Incidentally
by (4 . 1 . 7)
-=-&0 ..... (4.2.4)
n 2 v '
or the average energy of a molecule is 1-5 times kd.
4 . 3 Mixtures of Gases. This identification of the distri-
bution-constant with the inverse temperature is further
confirmed by studying statistically the distribution in phase
of molecules of different kinds in a vessel.
Let us deal with a mixture of two gases, n molecules of one
and n' molecules of the other. Consider a state in which
n l9 n 2 , ...... , n c molecules of the first gas are in the c
phase-cells ; n\, n' 2 , ...... , ri c of the second also in the
same phase-cells. Now such a state can be produced by
associating any complexion of the (n l9 n 2 , ...... , n c ) state
of the first gas with any complexion of the (n' lt n' 2 , ...... ,
ri c ) state of the second. Thus the number of complexions
embraced by the (n lf n 2 , ...... , n c , n' v ri 2 , ..... , ri c )
state of the mixture is the product of the separate com-
plexion-numbers for each gas. That is
W (n lt ?i 2 , ...... n e9 n' l9 n' 2 , ...... n' e ) =
n\ ___ ri\
n \ n 2 l ...... n c \ n\l
50 STATISTICAL MECHANICS FOR STUDENTS
and by the Stirling approximation
log W = n log n + n' log n' J Sn T log n, + Sn' t log n' r j
(4.3.1)
To find the normal state we must determine the values of
the n r and ri r , giving a maximum value to log W subject to
the conditions
Sn r =n . . . . (4.3.2)
Zn' r = n' . . . . (4.3.3)
and 27<r r n r + ZV r n' T = E . . . . (4.3.4)
where E is the energy of the mixture. Careful attention
should be paid to the form of condition (4.3.4). We have
not two energy conditions
27e r n r = constant
2V r n' r = constant
for the energy of any one component of the mixture does not
remain constant. The intennolccular collisions involve a
perpetual exchange of energy between the molecules irre-
spective of the group to which they belong, and so the energy
condition is expressed in one equation not two. It is this
feature of the equations which is responsible for the important
result which we shall deduce presently. (There is in general
no equality between the energy of one type of molecule in
a given phase and that of the other type in the same phase
since
2 I ~ 2 I (2
r ' r r .
r ~ 2m
and , _ r 2 + i)* + C 2
r 2^
quite apart from considerations of potential energy.)
On proceeding to- the solution of the problem on the same
lines as before, we find we have to satisfy
2(1+ log n r ) 8n r + Z (1 + log n' r ) 8< -
8n r =0
2Bn/ -
E T 8n r + 2V Src/ =
TEMPERATURE AND DISTRIBUTION CONSTANT 51
Using the method of undetermined multipliers we multiply
the second equation by A, the third by A', and the fourth by
//, and add. The 2c + 3 quantities, viz., the " normal "
values of the n r and n r ' and A, A', p are determined from
(4.3.2), (4.3.3), (4.3. 4) and the 2c equations
log n r + A + fie, = . . (4.3.5)
log n,' + A' + /**/ - . . (4.3.6)
(r = 1, 2, ...... c consecutively).
It is to be noted that, for the reason mentioned, the same
fji occurs in the c equations (4.3. 5), as in (4 . 3 . 6), although
not the same A. These equations yield the normal values,
n r = v r and n/ = v r ' where
C being e~* and C', e~ v ; C, C' and //, are, of course,
worked out in detail in terms of E, n, ri , and the parameters
involved in the r arid / by means of (4.3.2), (4.3. 3),
(4.3.4).
Thus the presence of one gas docs not upset the nature of
the normal distribution of the other, and, as a significant
fact, the same value of distribution-constant appears in the
normal state of each part. The methods of sections (4.1)
and (4 . 2) are once more applicable, and we find that the
average energy of any molecule irrespective of type is as
before, 1-5 /z," 1 and the temperature is (k^)" 1 .
If the two molecular systems were in separate enclosures
they would for an extremely large part of their history be
distributed in or near the normal state, but not necessarily
with the same distribution-constant. However, on mixing
them, their normal state now involves a common-distribu-
tion-constant. This statistical deduction, which as we have
seen is derived from the dynamical principle of the con-
servation of energy, is the analogue of the attainment of a
common temperature by two gases on mixing. The statis-
tical result can clearly be extended to a gaseous mixture of
any number of different gases.
In this connection Avogadro's hypothesis can be readily
deduced from these results. Since the two gases at the same
E2
52 STATISTICAL MECHANICS FOR STUDENTS
temperature have the same distribution-constant for their
normal states, it appears from (4.2.3) that they have the
same k. Therefore, the gas constants, R, for two quantities
of each gas, each quantity containing the same number of
molecules, have equal values. But this amounts to saying
that equal volumes of two gases at the same temperature
and pressure contain equal numbers of molecules. For a
gram -molecule of any gas, R is known to have the value
8 32 x 10 7 ; and the number of molecules in a gram-
molecule is known to be 6-06 X 10 23 . Hence the value of k
the gas-constant per molecule, is 1-37 X 10~ 16 .
From (4.2.4)
and so the specific heat of a monatomic gas at constant
volume is 3 R/2.
4 . 4. Potential Energy. If an external field of force (other
than the one postulated to limit the size of the gas) is acting
on the molecules, the integrations carried out in section
(4.1) require some little modification, since e is now the sum
of (f 2 + f] 2 + 2 )/2m and a function $ (x, y, z), and the
equations (4.1.3) and (4.1.4) become
y = n
^ (4.4.1)
D[ ..... f <f> e-^dx dy dtdt d, dC
exp
cZefydC = E . . . (4.4.2)
The first integral in (4 . 4 . 2) is the potential energy of the
whole gas in the external field of force. Call this <i>. Also
represent the integral
me
cfe
throughout the whole volume of the gas by the symbol F.
Then in (4 . 1 . 5) and (4 . 1 . 6) we replace D by DF and E
by E 4> ; so (4 . 1 . 7) is replaced by
TEMPERATURE AND DISTRIBUTION CONSTANT 53
2 (E - <&)'
so that /i" 1 is still equal to two-thirds of the average kinetic
energy of a molecule and the relation between the tempera-
ture and the average kinetic energy is not disturbed. The
existence of the field does not affect the distribution of the
velocities among the molecules ; it does, however, produce
lack of uniformity in the density of molecular concentration,
since the density in a small volume surrounding the point
(x, y, z) is proportional to
and thus decreases as we move to places of higher potential.
The reader should rightly appreciate these statements. In
a region at high potential there are not, of course, in the
normal state, as many molecules within certain limits of
velocity as in an equal sized region at a low potential ; but
the ratio of the numbers of molecules within two defined
extensions-in-velocity in a given volume is unaltered by the
field of force.
4 . 5 Some Remarks on Pressure. The method of calcu-
lating the pressure in section (4 . 2) is probably familiar to
the reader, who has no doubt already met it in some text-
book of physics. Nevertheless some further discussion of it
may not be out of place here, although it might have pro-
duced a rather long digression in section (4.2) from the main
object of the chapter.
It should be realised, of course, that the transference of
normal momentum is not to be considered as only from one
side of the element of area A to the other ; that would only
have given half the pressure. Calling the two sides of the
area a and 6, we have to estimate the components of
momentum normal to the area towards 6 transferred from
side a to side b per unit time, and the components towards a
transferred from b to a. These two parts are of course equal
in statistical equilibrium, and their sum is the pressure.
The procedure can be related very easily to the usual defi-
nition of pressure as force per unit area ; for if we conceive
a physical surface situated in the interior of the gas, molecules
54 STATISTICAL MECHANICS FOR STUDENTS
moving up to the surface and those which have just re-
bounded from it correspond to the two streams of molecules
for an element of area parallel and near to the surface, and
the algebraic change of momentum in a given time pro-
duced by the surface is the arithmetical sum of the normal
momentum -components towards the surface for the on-
coming molecules and normal components away from the
surface for the receding molecules.
In view of similar considerations arising later in con-
nection with liquids, it is very necessary to be quite clear in
each particular illustration as regards the meaning to be
attached to the word " pressure/' Besides the idea of rate
of transference of normal momentum, there is the familiar
notion of " hydrostatic pressure " as force per unit area on
a surface arising from a " body " force, such as gravity
being exerted on each element of the fluid. It should be
obvious that so far we have not had much to do with that
conception. In any case if it is a question of a relatively
small quantity of enclosed gas, its weight is negligible in the
treatment of its pressure, since it merely produces a small
excess in the pressure exerted by the bombarding molecules
on the lower part of the flask over that on the upper, because
of the slightly larger density. Of course in the treatment of
our atmosphere as a whole, the two ideas lead to the same
value ; for it is the earth's gravitational attraction which
holds our atmosphere to it and as the pressure (in the
" bombardment sense ") vanishes at its outskirts, its pressure
at lower levels is just equal to its hydrostatic pressure,
estimated in the elementary way as weight exerted on unit
area.
It is, however, in the case of a liquid that serious con-
fusion will arise unless care be taken. As we shall see later,
the notion of an " internal " or " intrinsic " pressure due to
transference of momentum across an element of area in the
interior arises in theoretical discussion just as in the case of
a gas ; but on account of the larger density this pressure is
relatively enormous, and we have no direct experience of it.
When we use the phrase " pressure of a liquid," we refer, as
a rule, to the hydrostatic pressure arising from its weight
TEMPERATURE AND DISTRIBUTION CONSTANT 55
and varying markedly with depth in the liquid. But as a
matter of fact we shall see later that in the statistical dis-
cussion of the liquid state the symbol p is not even used to
represent this magnitude either, but is rather associated
with the pressure of the saturated vapour of the liquid. In
due course we shall see why this is so, and also how it comes
that the relatively great internal pressure is not capable of
direct experimental detection.
There is in the case of a gas an interesting and instructive
way of connecting pressure regarded as rate of transference
of normal momentum per unit area and pressure regarded
as force per unit area. It will be recalled that, instead of
considering a physical boundary to a quantity of gas, we
imagined a field of force acting normally inward on the gas
molecules at its external layer. We also saw in the last
section that if </> (x, ?/, z) is the value of the potential energy
of a molecule in this field at the point (x, y, z), then the
density of the gas at this point is p where
P=A,e-'*
p being the uniform density of the gas throughout its
volume with the exception of the exterior layer referred to.
Now it is proved in text-books of hydromechanics that
if p stands for the pressure of a fluid in the hydrostatic
sense, then p is a function of position which is connected
with the body-force by the following equations
X=i??
p dx
Y =!?
pdy
Z=l^
p dz
where X, Y, Z are the components of the body force per unit
mass of the fluid. Since X, Y, Z vanish in the interior of the
gas, the hydrostatic pressure, p, is uniform throughout it.
In the layer, however,
X = - I 9 + etc.,
m ox _
56 STATISTICAL MECHANICS FOR STUDENTS
m being the mass of a molecule, and so
ty _ P ^
dx mdx
= _,
m ex
Hence
the constant of integration being zero since p vanishes as (f>
approaches infinity. Hence in the interior of the liquid,
where $ is zero,
P =J-
m p.
or n
pv = -
M
where n is the number of molecules in volume v. But this
is just the same equation as (4 . 2 . 1) where p, however,
represented pressure in the sense of rate of transference
momentum.
Equation (4.2.1) shows that if u 2 is the mean of the
squared velocities of the molecules, then
p = _pu 2
^ 3^
since E = - 2 mu* =p vu*.
2 2i
Thus the " mean squared velocity " is equal to (3 #//>)*,
and so is of the same order of magnitude as the velocity of
sound through the gas ; for the latter is known to be
(i<p/p)* where K is the ratio of the specific heat at constant
pressure to that at constant volume, and is between 1 and
1-66 in value. For hydrogen at C., the mean squared
velocity is 1,840 metres per second, for oxygen 460 metres
per second, etc., varying in fact inversely as the square
root of the density under standard conditions or inversely
as the square root of the molecular weight.
CHAPTER V
EXTENSION TO MORE COMPLEX MOLECULES
5 . 1 Equipartition of Energy. Without the assistance of
general dynamical theory, any treatment of molecules which
are endowed with a structure must in the nature of things
be inadequate. However, the author's aim being to intro-
duce the reader to statistical results of general interest
with as little delay as possible, leaving a more thorough
treatment to a later stage, we shall make shift to consider
the case of complex molecules with the aid of a simple
model which has rendered yeoman service in the past
to students not too well equipped with mathematical
artillery.
Let us regard each molecule as containing within itself
" oscillators," i.e., particles attracted by elastic forces
towards centres, where they would remain in relative
equilibrium with respect to the general body of the molecule.
The restoring force is supposed in each case to be pro-
portional to the displacement of the oscillator from its
centre in the molecule and so the oscillation is harmonic.
Let q stand for the displacement ; the velocity of this dis-
placement, dq/dt, is represented by q and the acceleration,
d 2 q/dt 2 , by g. If a is the mass of the particle and bq the
restoring force, then the equation of motion is
aq + bq =
and the solution of this is
q = A sin (cot 6),
where co is equal to (b/a)* and is called the "pulsance"
(2 TT X frequency) of the motion. A is the amplitude of the
vibration, and 6 is the " epoch-angle " determined by the
fact that at times t = 0/eo, 0/co TT, 0/co 2 TT, etc. the
67
58 STATISTICAL MECHANICS FOR STUDENTS
displacement is zero. The kinetic energy is aq 2 and the
1 2
potential energy is bq 2 . The total energy is therefore
Z
-b&? sin 2 (a>t - 0) + i a A 2 a> 2 cos 2 (arf - 0),
2t 2i
or a A 2 a> 2 . The momentum aq we shall represent by the
Zi
symbol p.
The amplitude and the epoch-angle are not determined
by the equation of motion ; they depend on the so-called
initial conditions ; e.g., in the case of a simple pendulum,
while the time of swing is determined by the length and
intensity of gravity, the amplitude and the actual instant at
which the string has an assigned inclination are arbitrary.
A and 6 are, in fact, two arbitrary integration constants
which enter the solution during the integration of the
equation of motion. Physically this means, as regards the
oscillator, that A and have definite values during a free
path of the molecule which contains it, but are altered at
every encounter between this molecule and any other. In
any free path of a molecule the internal oscillator is engaged
in a harmonic oscillation with a definite pulsance cu, but
with an amplitude and epoch-angle which vari^s^g^Pfeee
path to free path. To make this more obvious we may
consider a molecule as a frame of reference for the oscillator.
The set of axes with reference to which we estimate q are
fixed in the molecule. (For the moment we are regarding
the oscillator as having one " degree of freedom/' i.e.,
vibrating to and fro parallel to one axis). As long as the
frame of reference is moving with a uniform velocity, the
oscillation is not interfered with ; at time t
q = A sin (a>t 6)
q = Ao> cos (cut 6).
An encounter produces an acceleration in the molecular
frame of reference for a brief time. This is equivalent to a
force acting on the oscillator for the same time in the
EXTENSION TO MORE COMPLEX MOLECULES 59
opposite direction. We are here appealing to the well known
mechanical aspect of the relativity principle. We have only
to consider our experiences in a carriage, which has been
moving steadily and is then suddenly accelerated or retarded,
in order to realise the situation. The oscillator suddenly
receives an impulse in a direction opposite to that on the
molecule. When it is over, the state of affairs is such that
the oscillator has practically the same q but a different q
and for the subsequent spell of harmonic motion before the
next shock, q and q cannot be given by the equations above,
but by two such as
q = A' sin (a>t 6')
q = coA' cos (cut 0')
with a different A and 9, but, of course, with the same
pulsance as before. Hence at every collision not only is the
kinetic energy of the molecule changed, but also the internal
energy of the oscillator also. The collisions, in short, effect
an exchange of energy of translation between the molecules
and also between this energy and the internal energies of the
molecules.
We can as before bring this new " degree of freedom "
within our considerations of probability. The phase of a
molecule is now determined by eight components, x, y, z, q>
> ??> C, p, and an extension-in-phase by the limits x to x +
8x, , q to q + S#, to + Sf , , p to p + Sp.
The phase-diagram is now an eight-dimensional one ; or if
it is preferred, it can be visually represented by four plane
diagrams, a point in the fourth one representing q and p.
A phase is represented by a " point " in the phase-diagram
or a four-point group in the four plane diagrams. A phase-
cell can be pictured if one likes, as a group of four elementary
rectangles and a " path " as a group of four curves.
If now one " aspect " of a molecule is as possible as any
other with this extended notion of aspect, the probability
of a state in which n^ representative points are in the first
cell, etc., is as before given by the quotient of W (n l9 n%>
, n c ) by the total number of complexions consistent
with the energy condition. The procedure is just as before ;
60 STATISTICAL MECHANICS FOR STUDENTS
the state of maximum probability is given by equations
(3 . 2 . 1), (3 . 2 . 2), (3 . 2 . 3), where e is defined to be
2 + ^ + C 2 , P 2 , , , x , 1 L 9
! - \ I - \ - -J- _ -j- J) (x ,y , z) + - fr? 2 ,
2m 2a y v J J 2 *'
and involves the phase (<?, ^) and parameters (a, 6) of the
internal motion. The same type of proof also serves to
show that this state and those very near it constitute the
normal state of the system of molecules.
5 . 2 Partition of Energy. On reverting to the procedure
of section (4 . 1), we can easily obtain average values for the
various parts of the kinetic energy of a molecule which are,
as we say, associated with its various degrees of freedom.
In the first place the equation (4 . 1 . 3) is replaced by
D[ ..... fe"^ Ax dy dz dq d drj d( dp = n . (5.2.1)
In the absence of an external field this reduces to
e~w* dp = n
where the integrations are practically from GO to + oo
in each case * and
a=A)8=!/*&, y = -^.
2m P 2 P ' r 2a
We thus have
The kinetic energy associated with the motion of a
molecule parallel to the axis of x is 2 /2ra, and so the average
of this over every molecule at any moment is
~" d * ...... dp
divided by n ; i.e.,
^ j> e-* 1 dt . Je-" f dr, ...... Je-^ 1 dp
* See Note 1 to this chapter.
EXTENSION TO MORE COMPLEX MOLECULES 61
divided by the expression on the left-hand side of (5.2. 2).
This is equal to
2m 2
1
4 m a
1
A similar result follows for the average of ?? 2 /2ra and
and also for the average of the part of the kinetic energy
p 2 /2a associated with the internal movement.
The reader should carefully note the meaning of the word
" average " in this connection. No statement is made about
the average total or partial energies of an individual mole-
cule over a finite lapse of time. We cannot follow individual
histories. The " equipartition law " is concerned with
energies averaged over all the molecules at any one instant.
The modification introduced into the treatment of pressure
in section (4 . 2) owing to the considerations of internal
structure is easily dealt with. It is not difficult to see that
the pressure is now two-thirds of the density of the kinetic
energy of translation of the molecules (not of the whole
kinetic energy) ; so by the result just obtained
3
n
71
" r>
2,*
*
3 v
n
or pv = -
V*
where p is the pressure. As before, this result leads to the
* p is here the pressure and must not be confused with the momentum
of an oscillator.
62 STATISTICAL MECHANICS FOR STUDENTS
identification of the distribution constant p with (k6)~ l ,
where k is the gas constant per molecule.
Indeed the reader should have no difficulty in seeing that
the line of proof used can be extended to molecules with
oscillators having more than one degree of freedom, i.e.,
free to vibrate in all directions with reference to axes fixed
in the molecule with three degrees. Furthermore, molecules
containing more than one oscillator, each oscillator having
its own distinctive mass and force-constant, can also be
brought within the ambit of the proof. And lastly mixtures
of different molecules can be treated in a similar manner to
that used in section (4. 3).* The striking feature is the
appearance of the same distribution constant in the exponen-
tial factor of the distribution law, and this as we have seen,
is the result of the liberty of exchange of energy between all
the degrees of freedom of the molecules as a whole and of
the internal oscillators. The equality between /x" 1 ad kO
is still maintained and the average kinetic energy associated
with any degree of freedom is as before | kO. If p, p' ', p" ',
are the partial pressures of the constituent gases,
then
n
pv -- nkv
P
p'v = n'kB
etc.
where n, ri, are the numbers of each type of mole-
cule present, this being the familiar law of partial pressures.
It is very necessary to observe that the equipartition law
has been confined to average kinetic energies, and it is easy
to see that the restriction is connected with the fact that the
kinetic energy of each molecule is the sum of terms each of
which depends on the square of a momentum, involving in the
proof the possibility of splitting a certain part of the expres-
sion e~** into separate factors of the type e~ a *\ ,
e"^. It is true that this is just as much a feature of the
potential energy of the oscillator in a molecule, which is
given by a term involving the square of a co-ordinate ; and
* See Note 2 at the end of the chapter.
EXTENSION TO MORE COMPLEX MOLECULES 63
we can prove just as before that the average potential
energy of a harmonic oscillator is also ^ kd for each degree
of freedom. But in general potential energy is not given by
simple square terms. If the restoring force on the oscillator
were not proportional to q, the oscillator would be " anhar-
monic," and the potential energy would not be represented
by \ bq 2 and the equipartition proof would fail for it. Also,
as regards the external field of force, the potential energy
function <f> (x, y, z) is not in general a sum of squares and,
again the proof would fail in this connection. For example,
if it were a uniform field parallel to the axis of x, the potential
energy would be proportional to the -co-ordinate (choosing
the origin at a suitable level) and writing Ax for cf> (x, y, z)
we would obtain the average potential energy by dividing
DA f x e-^ Ax dx( (e~^' dy dz dq d dj] dC dp
by .
D l c"^^ I \e~^' dy dz dq d drj d dp
Jo J J
where e' = e Ax.
This is
r
which is equal to
2
i.e., juT 1 or kd, just twice the average kinetic energy. In fact,
if the potential energy were proportional to any power of
x, say x n , then it can be easily deduced that the average
potential energy would be k6/n. As a general rule, even
such relatively simple expressions as sums of powers of
the co-ordinates do not hold sway, and so the reader must
be careful in the use of the equipartition law, unless applied
to kinetic energy. Later, when we reach a fuller dynamical
treatment, we shall see that there is a general partition law
which covers the equipartition of kinetic energy and gives
64 STATISTICAL MECHANICS FOR STUDENTS
us some idea as to procedure in other eases. All this is, of
course, based on classical dynamical laws. Wider con-
siderations involving the quantum hypothesis will modify
even this general law.
We cannot leave the subject of complex molecules even
at this stage without some reference to molecules involving
more than one atom. A familiar picture for certain diatomic
molecules is that of a dumbbell two separate atoms held
rigidly together. This picture introduces new degrees of
freedom connected with rotation as distinct from trans-
lation. The student may naturally remark that rotation is
just as much a possibility for monatomic molecules as for a
diatomic and should therefore have been considered earlier.
The answer to this is to draw the reader's attention to an
implicit assumption in the model of a monatomic molecule
used hitherto. It has been regarded as dynamically equi-
valent to a hard smooth sphere. The mutual forces exerted
at encounters are normal to the surface, and passing through
the centre produce no change in rotational momentimi nor
in the individual rotational energies of the molecules. Thus
rotational energy is not exchanged between molecule and
molecule at an encounter, and since it therefore makes its
appearance in the equation (3 . 1 . 2) as a constant, it docs not
appear at all in (3 . 2 . 8). But it is otherwise for a dumbbell-
shaped particle ; the forces between the atoms of different
diatomic molecules will not produce any change in the com-
ponent of rotational momentum around the axis of figure,
but will do so about any axis at right angle to the figure-
axis. This amounts to excluding one of the three new degrees
of freedom.* Again we must anticipate later dynamical
work when we state that each effective degree brings in on
the average ^ k9 of kinetic energy of rotation. Of course, for
a less symmetrical diatomic molecule or for a polyatomic
molecule, provided we can regard it as rigid, we should have
to take into account three degrees of freedom for rotation.
These remarks have an important bearing on the specific
* The reader is reminded that rotational velocity about any axis can
be resolved into three component angular velocities about threo Cartesian
axes of reference.
EXTENSION TO MORE COMPLEX MOLECULES 65
heat of a gas. If the gas be monatomic, with no internal
degrees of freedom, the energy -content will be 1*5 nkff, and
so the heat capacity at constant volume will be 3R/2 where R
is the gas-constant for the amount of gas considered. This
is in good agreement with facts. For a diatomic gas we
would expect kO more energy on the average in a molecule,
and this would lead to 5R/2 as the heat-capacity. For more
complex rigid molecules we would expect a still further \ k0
of energy per molecule involving 3R as the value of the heat-
capacity. Provided temperatures are not too low or too
high, several gases show good agreement with these results.
Thus from elementary thermodynamical reasoning we know
that the heat-capacity of a quantity of gas at constant
pressure is greater by R than the heat-capacity at constant
volume, and so the ratio of the two specific heats for diatomic
gases should be 7/5 or 1-4, and for more complex gases 4/3
or 1*33, two results which are in good agreement with the
facts for some gases. But as diatomic gases are reduced in
temperature, it is found that their specific heat per gram-
molecule falls asymptotically to the values for monatomic
gases, from which it would appear t* t for some reason not
evident in the classical dynamical i , vtment, the rotational
energy has on the average a value p jressively smaller and
smaller than the amount kd per molecule. This is a dis-
crepancy which, as we shall see later, involves the use of the
quantum hypothesis to remove. At high temperatures the
specific heats show signs of attaining higher values than
those theoretically deduced. To account for this we have
to abandon the restriction as to rigidity in the molecule and
admit that the molecule may have kinetic energy arising
from the relative vibrations of its constituent atoms and
potential energy arising from relative displacements. But
again we have the same difficulty to meet. The classical
treatment does not show why these energies of internal
vibration and strain should not make their appearance at
normal temperatures. So long as elastic strain of the parts
of the molecule are postulated the relative partition of the
total energy among the various degrees of freedom is settled
not by the particular value of p, but by the expression for
66 STATISTICAL MECHANICS FOR STUDENTS
the energy of a molecule in terms of its co-ordinates and
momenta (external and internal). Thus for all " squared
terms " there is partition on an equality basis ; and for
other terms though not so simple, it is definite. Nor is this
all. The facts of the spectral lines of gases show that there
is a quite complicated vibrational mechanism within every
atom and molecule which is the dynamical equivalent of
numerous oscillators. Why then should the specific heat of
any gas not be very much greater than it really is ? On
any reasonable assumption in addition to the half-dozen or
so energy-quantities of |- TcO assigned to each molecule, there
should be many more for the internal degrees of freedom of
the radiating mechanism. Again these discrepancies can
only be dealt with by an appeal to some form of quantum
theory and there we must leave it for the present.
NOTE 1. Since the oscillator must in its vibration be
confined within the molecule, it may appear absurd to
integrate for q from oo to + oo. Actually it is not incon-
sistent with our knowledge of the sizes of atoms and molecules
to conceive that a displacement of the oscillator is quite
possible which renders the potential energy very much
greater than the average value (2 j^)" 1 ; for such a displace-
ment e~ M * would have reached such a minute value that
the part of the integral between this value #nd infinity is
negligible. A similar remark applies to the p integration.
NOTE 2. A slight difficulty in the mathematics may
present itself to the reader in connection with the fact that
the number of oscillators or degrees of freedom within a
molecule of one type in the mixture may not be the same as
the number within a molecule of another type ; so that we
cannot apparently use a common set of phase-cells for each
group of molecules, the dimensionality being different in
EXTENSION TO MORE COMPLEX MOLECULES 67
each case. Formally we surmount the difficulty easily by
bringing the number of oscillators in every molecule up to
the same value. We can then conceive the " extra " oscil-
lators in those molecules which have really less than the
maximum number to possess zero mass and so to contribute
no energy to the total amount.
?2
CHAPTER VI
THE SECOND LAW OF THERMODYNAMICS
6 . 1 The Normal State of a Molecular System and Thermo-
dynamic Equilibrium. In accordance with our aim of
bringing out the connection between statistical mechanics
and thermodynamical facts as soon as possible, we propose
in this chapter to deduce the second law of thermodynamics
for a gas from the results hitherto obtained, before proceed-
ing to a further development of the subject itself.
We have seen that the molecular system is, if our prob-
ability-postulate be accepted, for a relatively great part of
its history in or extremely near to the " normal " state
defined by the c equations
Vf = C e"*"'
or
log V r = A fJL r (6.1.1)
where A is written for log C. (For convenience a change of
sign in A from the earlier sections is made.) These, combined
with
v=l
c
. . . . (6.1.2)
Z r v r = E (6.1.3)
serve to determine, the v r > A and //, as functions of E, n and
the parameters which we shall denote by the symbols
a l9 2 , . . . . , a e . Let us change our point of view a little and
regard (6.1.1), (6.1.2) and (6 . 1 . 3) as equations deter-
mining the v r , E and A as functions of /z and the parameters.
That is, we are going to consider the molecular system
passing from the normal state for one value of ^ to the normal
68
THE SECOND LAW OF THERMODYNAMICS 69
state for another value of ft, involving, of course, changes in
the energy E, as well as in A and the individual v r or normal
numbers in each phase-cell. Remember that p has no
meaning for the molecular system apart from the normal
state. If the system be not in the normal state, it is a direct
result of our postulate that it will gradually tend to it. There
is no dynamical impossibility involved in a statement that
there might be abnormal states of distribution in which the
system might remain for ever. All we can say is that on our
probability basis it is enormously improbable. It is not
impossible that all the peoples of the world could be on the
Isle of Man in one day (actually a density of one person per
square yard would just about suffice) but well ! the point
need not be laboured.
The analogy between this normal state and the state of
thermodynamic equilibrium of an isolated physical system
is too obvious to escape notice, and the analogy is very close
indeed ; for from the equations of the normal state we can
construct functions of /z and the parameters which are as
a matter of pure mathematics connected by differential
equations of precisely the same form as those which are found
by experiment to connect those functions of the thermo-
dynamic variables of a system which we call the internal-
energy-function, the free-energy-function, and the entropy-
function. The proof which follows is, to be sure, limited at
the moment to gaseous systems ; but it is surprising how
little further elaboration of mathematical detail is required
when we deal with this question for other systems at a later
stage.
6 . 2 The Entropy Law. On solving (6 . 1 . 1) (6 . 1 . 2) (6 . 1 . 3)
we express the v r as functions of ft, a l9 a 2 , a e (remem-
bering that l e 2 , , c are functions of a v a 2 , ,
aj. Inserting these in (6 . 1 . 3) we obtain a function of ft, a x ,
2 > , # e > which is equal to E ; we shall denote it by
H (/x, a v a 2 , , a e )
or briefly by
H (p, a).
Also from (6.1.1) we express A as a function of the ft, and the
70 STATISTICAL MECHANICS FOR STUDENTS
a r . Let us write for n A//H. the functional form ^P (/*, a v a 2 ,
...... , a e ), or
Suppose the molecular system to change from the normal
state for the distribution constant JJL and parameters a v a 2 ,
...... , a e (the statistical-mechanical (S.M.) variables) to
the normal state for values /U, + 8/i, a l + Ba l9 a 2 + 8a 8 ,
...... , a e + 8a e of the S.M. variables. The energy will
alter to a value E + SE where
8E = H (p + fy, a + 80) H (/*, a).
Of this change in the energy, a certain part is given by
c l oa 2
We shall denote it by SE l9 so that
. . (6.2.1)
This part arises from the change in the parameters, but
with the distribution still kept at the original normal state
(n r = v r ) and not altered to the new normal state (n r
v r + 8v f ). The remainder of the energy 8E 2 ( = SE SE X )
is given by
8E 2 = i;J:6 r ^8a, + i:c r ^s M . . . (6.2.2)*
r=l t i ca 9 r -l CIJLC
The first part, SE l9 does not arise from effective changes
in the co-ordinates and momenta of the molecules ; for the
numerical distribution of phases among the cells is regarded
as unaltered in estimating SE 1 . The part 8E 2 does arise
from the changes in cell-distribution occasioned by the
changes in the parameters and ^. It is plausible to regard
this second part as the analogue of the heat supplied to the
system and the first part as the analogue of energy trans-
ferred to the system by purely mechanical means. We recall
the fact that among the parameters are a group related to
* Remember that the ( r do not depend on ,u.
THE SECOND LAW OF THERMODYNAMICS 71
an external field of force which limits the volume of the
molecular system ; any change in these involves alteration
in the boundary and work of an " external pressure." Thus
SEj may be regarded as the analogue of " external work "
done on the system.
Quite apart from such interpretation, however, it can be
deduced as a matter of mathematics only that, with SEj
and SE 2 defined as in (6 .1. 1) and (6 .1, 2),
/.SE 2 = 8{^[H(^a)-^( M ,a)]} . . . (6.2.3)
We shall defer the actual mathematical steps for a moment
so as not to interrupt the general line of thought. Suppose
the molecular system to experience a finite change from a
normal state (p,', a') to a normal state (//,", a") passing through
all the intermediate normal states on its way (just as in thermo-
dynamic reasoning a physical system is supposed to pass
from state to state through intermediate states of thermo-
dynamic equilibrium), then
= //[H (//, a")- V (,/, a")] - / [H (//, a') - ^ (//,a')].
Thus the integral of JJL d E 2 from the state s' to state s" along
a track of normal states depends only on the initial and final
states. The analogy with the entropy-theorem of thermo-
dynamics is obvious. On interpreting 8E 2 as heat supplied
to the system, and (kfji)~ l as the temperature 0, we have
> (ju/, a')
J 9 > U
where
and is the analogue of the entropy -function of the thermo-
dynamic state of the physical system.
To complete the analogy we have to discover the statis-
tical-mechanical theorem corresponding to the increase of
entropy which takes place when an isolated physical system
passes in an irreversible manner from one state of equilibrium
to another. Such an irreversible change takes place through
intermediate states some of which at least are not states of
72 STATISTICAL MECHANICS FOR STUDENTS
equilibrium. Hence in the statistical-mechanical analogue
we must conceive of some way in which a molecular system
in a normal state may pass to another normal state through
intermediate states which are not all normal. This can easily
be effected. In the thermodynamic processes we alter the
thermodynamic variables from the values which hold for the
first state to those which hold for the second in a short time.
This ensures irreversibility of the path. The essence of
reversibility is the infinitely slow change in the variables
allowing the system to pass through intermediate states of
equilibrium. Similarly we quickly change the S.M. variables
from the values //, a' for the first state to the values /z", a"
for the second state. To carry through the reasoning we
require to know another mathematical result, the proof of
which we shall also defer for a moment ; it is this :
3> (p, a) = k (log W m - n log n) . . (6.2.5)
where W m is the maximum value of the function W (n l9 n^
n c ) being equal in fact to W fa, v 2 , v c ).
Suppose then the molecular system is in the normal state
characterised by (//, a\, a' 2 , , a' e ) ; a sudden change
is made in the S.M. variables to the values //, a'\, a" 2 ,
. . . .v. . , a" e .* The distribution v/, i> 2 ', v c ' is no
longer the normal distribution for the new variables ; the
normal distribution is i//', j/ 2 ", , i/' c .
Hence
W fa', iV, v/)< W fa', /, c ")
for the latter is the maximum value of the function W (n v n 2 ,
n c ) for the new parameters and distribution-constant.
At the beginning of the change the system although in a
normal state for the old S.M. variable is in an abnormal state
for the new S.M. variables, and by our probability-postulate
it will ultimately arrive at and keep very near to the normal
state for these new values. It may experience all sorts of
changes on the way ; W may fluctuate about in the most
fortuitous way (we cannot follow the individual history of
* Remember that there is no change of energy. The system is to be
isolated. Hence changing parameters will in general involve a change of
M also, so as to satisfy H (/*', a') = H (M*, *)
THE SECOND LAW OF THERMODYNAMICS 73
each particle), but presently the system will reach the new
normal state with the increased value of W. If W m ' is the
value of W (v 1 / , v 2 ', ...... , v e ') and W m " is the value of
W (y/, v \ ...... , v c ") 9 W m ' was the maximum value for
the S.M. variables in first state, just as W m " is for the
second ; and since W m ' < W m ", therefore by (6.2.5)
Thus in the statistical-mechanical argument the function
4> (/x, a) behaves in a manner similar to the mode of behaviour
of entropy in Thermodynamics.
We have to fill in the missing mathematical steps leading
to (6.2.3) and (6.2.5). They are three in number.
I. Since any one of the equations (6 . 1 . 1) is an identity
when the v r , X and the e r are regarded as functions of /* and
the a,, the result of partial differentiation with respect to /z
or any of the a, still yields identities. Hence we have the c
equations
3 log v r __ 3A ___
for any given a,.
This is the same as
__ ___
da s Sa 8
dv r __ d\ _ 3e r
da t r da 8 r da s
On^adding the c equations we obtain
dn dX c) r
~n~ ^S v r ~^.
ca 8 ca s r =i va g
But, since n does not change, dn/da, = 0, and so
Sv 8g r^ 8 ^(^ q ),
r i f da g da s
Thus
Z v ***a. = S* lf <>'.*} Sa
,_i ,=! da, ,,i da,
But
S* 0*. a) - -^ V 4, j?^0*.) 8a
74 STATISTICAL MECHANICS FOR STUDENTS
Hence
SEi = 8V OK, a) ^P-^-' 8/i.
As 8E 2 = SE - SE X it follows that
SE 2 = SH (u, a) S^ (u, a) + ^-^ 8/t (6.2.6)
3/A
II. Reverting to (6.1.1) once more, we obtain the c
equations
9 log v r 3A
5 == 5~ r>
C//X. C7/X,
or 3v r _ 3A
which on addition yield the result
So that
CfJ,
or
_ _ (6.2.7)
a/* ^
Combining (6.2.6) and (6.2. 7), we have the result
8E 2 = 8[H 0., a) - * (M, ] + H ^a)-^0*.a)
or ^ 8E 2 = 8 { /i [H (^ a) - ^ (/i, a] j ,
which is equation (6.3. 3).
III. Lastly, on putting the values for v r in the expression
for log W, we obtain
c
log W m = ft log n 27 v f (A p r )
ri
= n log 7i /&A + /z E.
So
/i [H (/i, a) ^ (/i, a)] = log W m - n log w,
THE SECOND LAW OF THERMODYNAMICS 75
and
<E) fa, a) = k (log W m n log n),
which is equation (6.2.5).
The molecular models which we employ thus give us a
somewhat broader view of the property which we call
" entropy " than pure Thermodynamics. The function
W (n l9 n a , , n c ), is one which does not maintain a
constant value in the actual history of a molecular system ;
there are continual fluctuations in its value going on as the
numbers n l9 n 2 , , n c change with the individual
movements and encounters of the molecules. The probable
amounts of these fluctuations we shall estimate later, but
they are on the average small, although there is no dyna-
mical impossibility in brief excursions to values very far
removed from the normal v f . So if we adopt the terminology
of Darwin and Fowler, and call
A | log W (n l9 n 2 , , n c ) n log nj ,
the " kinetic entropy " of the system, we know that this
kinetic entropy is fluctuating between its maximum value
and values slightly below it, except for very occasional wider
fluctuations to values well below. In thermodynamics,
which is built on " macroscopic " observation, not " micro-
scopic " analysis, we rest our reasoning on experimental
results in which these fluctuations become " smoothed out,"
and an appearance of statical rest rather than of statistical
equilibrium is presented, leading to the conception of a
non-fluctuating thermodynamic entropy for whose value we
naturally take the maximum value of the kinetic entropy.
In thermodynamics, moreover, entropy is a property which
can only be defined for states of equilibrium. In Statistical
Mechanics, kinetic entropy is not so restricted to normal
states alone.
To conclude this chapter we have to point out that the
function (/*, a) is the analogue of the free-energy function
(at constant volume) of a thermodynamic system. Thus
(6.2.4) can be written
9 =H - 0*,
76 STATISTICAL MECHANICS FOR STUDENTS
and, moreover,
dp,
Id0
/ dp
which, by (6 . 2 . 7) = fyt (H ^)
= 4>.
These two results connecting W (/x, a) with H (//,, a) and
4> (/x, a), are obviously formally similar to the mathematical
equations connecting the free-energy function with the
energy and entropy functions in Thermodynamics.
CHAPTER VII
THE ENTBOPY OF A PERFECT GAS
7 . 1 The Problem of the Magnitude of a Phase-Cell. In
Chapter IV. we carried through some mathematical opera-
tions which involved the substitution of integrations of con-
tinuous functions of phase for summations of a number of
terms of a series. The nature of the problems which we were
discussing at that point is such that no ambiguity results
from this procedure. It is quite otherwise with the investiga-
tion on which we are now about to fasten attention. In the
immediately preceding sections we have obtained an expres-
sion for the entropy of a monatomic gas ; it is
* (log w m n log n)
c
or k 2 v r log v r .
r=i
Naturally we are concerned to discover the connection of
this expression with the well-known thermodynamic expres-
sion for the entropy of a gas, viz.,
s p log R log p + constant
where s p is the heat-capacity at constant pressure. The two
mathematical expressions do not bear a very obvious resem-
blance to one another ; yet the conversion of the former into
an integral leads quite directly after a few steps to a demon-
stration of their essential agreement. But there is one step
in the conversion which has been the origin of one of the
most famous scientific discussions of the past two decades,
and has led to results which are in some quarters regarded
as one of the prime achievements of the quantum theory
outside its triumphs in settling questions of atomic structure.
In Chapter IV. we replace n r by a sextuple integral
f f
77
78 STATISTICAL MECHANICS FOR STUDENTS
where da is written as a symbol for the infinitesimal element
of phase-extension
d dr\ dCdx dy dz,
and the integration is extended over limits determined by
the r th phase -cell. Now we clearly cannot replace log n r by
J ...... J log/efo,
nor n r log n r by j ...... J / log / da.
Strictly n r log n r is the product of I ..... I / da and
log ...... / da, both integrations Extending over a
phase-cell. If we wish to replace n* r log n r by an integral
over the r th phase-cell, and not by a product of an integral
and its logarithm, we must begin by introducing a symbol
for the magnitude of the cell ; let it be g. Then n r /g is the
average value of the function/ over the phase-cell ; it is the
average density of the representative points in the cell. It
follows that there is an approximate equality between log
(n' r /g) and the value of log / at any phase of the cell, and so
there is an approximate equality between the integral
and the expression
or n r log n f n r log g.
It follows that there is an approximate equality between
c
Z n r log n r ,
and the expression
J J /log/ da + nlogg,
the integration being between the extreme limits of the
phase-diagram. For the normal state we have seen in
THE ENTROPY OF A PERFECT GAS 79
Chapter IV. that the form of the function / (ignoring any
external field of force) is
Dexp {-a^+^ + f 2 )}
where
a= -^ . ..... (7.1.1)
4mE v '
, T, n / 3n \ 3 / i \
and D = - ( - _ I . . . . (7.1.2)
v v
and so there is an approximate equality between
c
E v r log v r
r=l
and
X flog D - a (* + ij 8 +P)ldv+nlogg. . (7.1.3)
the integration extending between the widest limits of the
phase-diagram.
The quantity g is very vague at the moment. We only
know two things about it. Although not a mathematical
infinitesimal, it is " physically small " from the point of view
of the experimentalist ; yet it must be large enough to
contain very many representative points in the region of
the phase-diagram favoured by the points in the normal
state. Further its physical dimensions in terms of the
fundamental quantities length, time and mass are the same
as the cube of the quantity " action," which is defined as
the product of energy and time, and is a concept of great
importance in the general mathematical formulation of
classical dynamical principles ; for the product of energy
and time has clearly the same dimensions as the product of
momentum and length. Nothing more can be elicited about
g from classical sources.
It is more than twenty years since Planck made a very
definite proposal about its magnitude. He suggested that
there is a precise definite value for g which makes the equality
mentioned above not merely approximate but exact. Natu-
80 STATISTICAL MECHANICS FOR STUDENTS
rally such a statement could only be advanced with the
support of some physical facts hitherto unnoticed. The
facts which Planck appealed to were just those newly-
observed phenomena which were at the time leading to the
acceptance of a rather indefinite form of quantum hypo-
thesis with its then revolutionary idea of a kind of atomicity
in atomic occurrences previously unsuspected. There was
also a new development in purely thermodynamic theory
which was proving a fertile instrument of progress in the
hands of Nernst and his pupils. Naturally we cannot enter
into a discussion of these matters here, and must leave until
later chapters some account of the repercussions on statistical
mechanics of the quantum theory. But these remarks and
the mathematical analysis in the next page or so will serve
to prepare the reader for a fuller statement at a later stage.
He is already probably aware that one fundamental feature
of the quantum theory is the significance of a certain unit,
or, if you like, " atom," of action known as " Planck's
constant," and denoted by the symbol h. In terms of the
erg-second as a working unit of action, the value of h has
been determined in several ways to be 6'55 X 10~ 27 ; and
to cut the matter short at the moment, the upshot of a great
deal of discussion and experiment has been the suggestion
that g = A 3 , since, as pointed out above, g has the dimensions
of action cubed. Without taking any other step for the
present than that of using g to represent a precise physical
magnitude, let us proceed.
7. 2 The Entropy Constant. By (7.1.3) the entropy of the
gas becomes
2 + ? + C 2 ) exp{ - a (f 2 + T? 2 +
- AD log Df ...... f ea?p{- a (P + *? 2 + C 2 )}*r knlogg
The first term is simply 2 JfcamE, or, by (7 . 1 . 1), 3 nk/2.
The second term similarly reduces to n k log D. Thus the
entropy of the gas becomes
n k log D n k log g.
2
THE ENTROPY OF A PERFECT GAS 81
On inserting the value of D from (7.1. 2), and completing
a few obvious steps, we arrive at
7(3, -nil 5, i 3 /, . , 4 77 m\ , )
n k - log E + log v - log n + - ( I + log ) log g[
{Z * A \ O / J
Since E = 3 n k 0/2, we easily obtain for the entropy of
a monatomic gas containing n molecules the expression
7 ( 3 7 4 , , , , , (2 TT mkf . 3 )
ft k j - log + log v log n + log + - [ >
(2 g 2)
which, on changing from the variable v to p by means of
^w = R0 = nkO,
or log v = log log p + log ft- + log i
becomes
(5 (2 TT rafc) & 3 )
R \ log log p + log 1 > (7.2.1)
(2 2j
The formal similarity of (7 . 2 . 1) with the thermodynamic
expression for the entropy of a gas is apparent. In view of
Planck's hypothesis concerning the definiteness of the
magnitude of g, it goes further than pure thermodynamics,
for it suggests a precise value for the undetermined constant
of integration which enters in the thermodynamic analysis.
Now this is tantamount to asserting that a quantity of gas
has at a definite pressure and temperature an absolute
entropy, a position quite untenable in the classical thermo-
dynamics of the nineteenth century in which only differences
of entropy between two assigned states were considered.
But, as mentioned already, Nernst's heat theorem, and the
work carried out in his laboratory on measuring the affinities
of chemical reactions and the specific heats of gases and con-
densed materials at low temperatures, had in the early years
of this century carried physical chemists at all events well
away from the vagueness of the older position, even before
the full blast of the quantum theory had played havoc with
classical methods and conceptions. We cannot pursue this
particular matter further at the moment. The main problem
of this chapter has been to show the essential agreement
between two formally very different expressions, and that
has been achieved . We must pass on to further developments
of statistical theory along classical lines.
CHAPTER VIII
THE STATISTICAL THEORY OF CHEMICAL EQUILIBRIUM
IN A GAS REACTION
8 . 1 Reactions Equivalent to a Simple Dissociation. Let
us consider a system in which there exist atoms of two
different types and diatomic molecules, each containing one
atom of each type. The phase diagram represents the
Cartesian co-ordinates of the centres of gravity of the
dissociated atoms and of the molecules and the corresponding
momenta ; it is, of course, six dimensional. Let there be
present in all a atoms of one type and /? of the other ; so
that if the dissociated atoms at any moment are a and b of
each kind respectively, then the number of molecules at that
moment is I, where
a -f I =a /ft i n
b + l=p ..... (8.1.1)
It is assumed* that the formation of a molecule or its
dissociation is accompanied by the liberation or absorption
of a definite amount of energy. Thus, if there is no external
field of force, the energies of the atoms and the molecules in
a certain phase are given by
2m
.
2 ! ^2 I
__
2(m a + m b )
where w is a definite energy of dissociation and is regarded
as positive if energy is absorbed in the dissociation of a
molecule.
In counting complexions we have to analyse the situation
CHEMICAL EQUILIBRIUM IN A GAS REACTION 83
more exhaustively than in earlier chapters. Let us consider
the state in which
#! atoms of first type, &J atoms of second type,
/! molecules are in cell 1.
a 2 atoms of first type, 6 2 atoms of second type,
1 2 molecules are in cell 2.
a c atoms of first type, b c atoms of second type,
l c molecules are in cell c.
The first step begins by supposing that we have a par-
ticular atoms of the first type, b particular atoms of the second
type, and I particular molecules to deal with ; a, of course,
being Z a l r , b being 27 6 r , and I being 2 l r . There are, of
course,
aA
ways of distributing the first type atoms in the manner
indicated.
&!_ __
bjbj.. .. . .& c !
ways of distributing the second-type atoms ; and
ways of distributing the molecules. The product of these
three expressions is then the number of different ways of
producing the state indicated with the particular atoms and
molecules chosen. But the italicised phrase shows us that
we have by no means obtained all the complexions embraced
within the state indicated.
We have, in fact, to bear in mind that we can select a
atoms from the given total number, a, of first-type atoms in
or
a ! (a a) ! a ! l\
02
84 STATISTICAL MECHANICS FOR STUDENTS
ways, and b atoms from the /? atoms of the second type in
J*L
bill
ways. Hence the previous product must again be multi-
plied by these two factors since a change of the individuality
of a single atom in a cell yields a different complexion. Nor
is this all. Having selected the dissociated atoms of each
type, we have by no means settled the individuality of the
molecules although we have, by reason of the selection
mentioned, chosen the 2 I atoms out of which they are
to be constituted. Let us consider a definite complexion
obtained as indicated. Without altering the individualities
or numbers of the atoms in the cells, and without altering
the numbers of the molecules in the cells, we can obtain I !
complexions from this one complexion by permuting the
atoms in the molecules. It is just as if we had dumb-bells
each made with a red and a black ball arranged in a par-
ticular scheme. Imagine the red balls to be removable ;
there are I ! different ways of arranging them in the I places
of the scheme and attaching them to the fixed black balls.
Thus to obtain the total number of complexions consistent
with the state described, we have to multiply the product
of the previous five expressions by II. The final result is
_ ^^
(8.1.3)
The calculation of the most probable state proceeds as
before by taking the logarithm of this expression and varying
it under the assigned conditions of number and energy.
Using the Stirling approximation, we obtain the following
variational equations for the state of maximum probability
Z log a r 8a r + 2 log b r 8b r + Z log l r 8l r =
Z $a r + Z U r =
Z 8b r + 2 81, =
Z ar 8a, + Z br 86 r + Z lf $l r =
CHEMICAL EQUILIBRIUM IN A GAS REACTION 85
These, by the method of undetermined multipliers, give
us 3 c equations
log a r + A a + M *ar =
log b r + A 6 + fi c* = . . . .(8.1.4)
log l r + A a + A, + p, lr =
These equations combined with
are sufficient to determine the 3 c + 3 quantities, A fl , A 6 , p,
and the a r , b r) l r in terms of E, n and the parameters. The
usual feature of a common distribution constant appears
once more and in the normal state the numbers are
a r = A exp(-p. ar )
l r = A B
where A = exp(-)( a )
The well-known mass-law follows at once from this. For
the total number of atoms of the first type in the normal
state is given by an integral (after the manner of Chapter
IV.) such as
= -J ...... Jea?p(-fu: a )i
_ A /2m a 77\i
~~ g \ ft /
\dv
_B /2jr
flf \ ft
== AB /2(m. +
gr V /i
Thus, since the concentrations are proportional to the
n^mbers, we have the usual law
86 STATISTICAL MECHANICS FOR STUDENTS
where K is the equilibrium -constant depending on the
temperature and the energy. Further, since K oc e~* w
= -
R0 2 '
where Q is the heat of dissociation of a gram -molecule of the
molecular compound.
8 . 2 Reactions in which the Number of Molecules is Un-
changed. Turning to a reaction in which two molecules of
the type XY yield one molecule of the type X 2 and one of
the type Y 2 symbolised by
2 XY ^ X 2 + Y 2 ,
we have in a particular state I molecules of XY, a of X 2 and
b of Y 2 , so that if a is the total (and given) number of atoms
X, and j3 the total number of atoms Y present, then
2a " M==a (82 n
26+Z=j3 * ' ' (8 ' 2 - 1}
Each molecule is characterised by a definite amount of
internal potential energy over and above its kinetic energy
as a whole,* so that
4 m a
_e + ^ +
& w>h
where w a , w b , w t are definite amounts of energy, positive or
negative.
The counting of complexions proceeds on much the same
lines as before. Considering a state as defined in section^
* We are excluding rotational energy. This could be treated by fresh
co-ordinates and extending the dimensionality of the phase -diagram. It
would complicate the mathematics without adding anything vital to the
discussion at this point.
CHEMICAL EQUILIBRIUM IN A GAS REACTION 87
(8.1) we have, so long as we particularise the molecules, to
obtain the number of complexions by multiplying
a! 6! /!
But for any complexion we can select the 2 a atoms in the
X 2 -molecules in
a! a!
or
2 a ! (a 2 a !) 2 a !Z!
ways out of the a X-atoms ; and out of the /J Y-atoms we
select the 2 6 Y 2 -molecules in
_L
2b\l\
ways. As before, this does not exhaust the possibilities. In
any choice we are left with I particular X-atoms and I par-
ticular Y-atoms to make the I XY-molecules ; but out of
any complexion with particular XY-molecules we make I \
new complexions by keeping the X-atoms of these molecules
fixed, as it were, in the design and permuting the Y-atoms
in all possible ways ; for thus we form new molecules with
the same atoms as before. We must be careful in applying
a similar process to the X 2 -molecules. Out of any com-
plexion with a particular X 2 -inolecules, it would look as if
we could make 2a ! new complexions by permuting the 2a
atoms all possible ways in the design ; but this gives us too
large a result ; for many of these would only be replicas of
other complexions (in the same state) made with the same
a X 2 -molecules. Any permutation which did not destroy
the companionship of the pairs of atoms would do this, and
this fact shows us that the complete permutations suggested
would reproduce definite complexions over and over again
as many times as there are permutations of the a molecules
(each regarded as a unity) in the scheme. But this number
of times is a !, and so out of the particular complexion with
a particular X 2 -molecules, we can only produce 2 a I/a ! new
complexions by permutation of the atoms. A similar
remark applies to the Y 2 -molecules. Thus the product of
the five expressions quoted above must be multiplied by
88 STATISTICAL MECHANICS FOR STUDENTS
I !, 2 a I/a I and 26 !/6 ! to obtain the number of complexions
possible for the state mentioned. It is
ajaj ...... ailbjbjt ...... b c \ IJ. lj ...... l c \
as before. The normal state is obtained by the variational
equations of the type
Z log a\ 8a r + Z log b' r Sb r + 2 log l r 8l r =
2 Z 8a r + 2 8l r =
2 Z 8b r + 2 SZ; =
2 ar Sa r + 2 e br Sb r + Z er Sl r = 0,
leading to 3 c equations such as
log a r + 2 A a + n * ar -
log b r + 2 A, + p br =
log l r + (A fl + A 6 ) + p, e lr -
and three further equations depending on total numbers and
energy. Thus in the normal state
. . (8.2.3)
l r = ABexp(-p, lr )
where
A == exp(-\ a )
B = exp(-Xb)
This result, combined with (8.2. 2), leads directly to the
result that
& 6 ____ T7"
where K is the equilibrium constant and is proportional to
t* ( w a
a b
expression w a + w b 2 w t is the energy absorbed in
the reaction
2 XY -> X 2 + Y 2
and yielded in the reverse reaction, so as before
eZlogK_ Q
d8
where Q is the heat of reaction at temperature 6.
CHEMICAL EQUILIBRIUM IN A GAS REACTION 89
8 . 3 Generalisation for Any Type o! Reaction. The method
of procedure for any type of gas reaction is now fairly obvious.
Let there be in all a atoms of type X, j8 atoms of type Y,
y atoms of type Z, and so on. In the reaction there occur
molecules containing x l of the X-atoms, y l of the Y-atoms,
z I of the Z-atoms, etc., #i, y l9 z ly etc., being positive integers ;
these we shall call Li-molecules. There will also be La-mole-
cules, containing x 2 of the X-atoms, y 2 of the Y-atoms,
z 2 of the Z-atoms, etc., and so on. The reaction follows
some stoichiometric equation, such as
v l LI + v 2 L 2 + v 3 L 3 + ...... =0
where v x , i> 2 , v 3 , ...... are integers, some positive, some
negative. This being so, the following equations must be
true
v l yi + V 2 2/2 + "3 2/3 + ...... (8 . 3 . 1)
The energies of the molecules in a given phase of position
and translational momentum are for an Li-molecule
for an L-molecule
and so on, where w l9 w 2 , ...... are definite internal energies,
and m v m 2 , ...... are masses of the respective molecules.
The heat of reaction during the reaction of v Li-molecules,
v 2 L 2 -molecules, etc., is
*i w i + "2 W 2 + V 3 ^3 + ....... (8.3.3)
If now in any state there are present l^ Li-molecules, Z a
L^-molecules, etc., then
X l l l +x 2 l 2 + x 9 l 3 + ...... == a
yi'i + yi'i + y a 'a + ...... =P (8.3.4)
z l l l + z 2 l 2 +Z B 1 3 + ...... = y
90 STATISTICAL MECHANICS FOR STUDENTS
The number of complexions in which there are
l n Lj-molecules, J 21 L 2 -moleeules, Z al L 3 -molecules, ......
in cell 1
Z 12 Li-molecules, Z 22 L 2 -molecules, Z 32 L 3 -molecules, ......
in cell 2, etc.,
is, after a series of steps similar to those taken previously,
found to be
The variational equations are
-Slog ^ SZi, + log Z 2r 8Z 2r + ...... -
x l 2 81^ +x%Z 8l 2r + ...... =
Vl Z Sl lr + y z 2 8/ 2f + ...... =
+zZSl ...... -0
2 lf 8^ + 2 c 2r 8Z 2f + ...... -
leading to equations such as
I
2r
and thus we find that in the normal state the numbers of
each type of molecule in the r th cell are given by
l lr = A*> B* (? exp(- Me lr )
(8.3.6)
where
A = exp( A ), B = exp( A 6 ), ......
As before, integration over the phase-diagrams gives the
total number of each kind of molecule present in the normal
state, and we find for the separate concentrations
C 2 = M 2 A 35 - B^ (? ...... exp( - /W 2 ) . (8.3.7)
CHEMICAL EQUILIBRIUM IN A GAS REACTION 91
where Mj, M 2 , ...... are numerical multipliers arising in
the integrations. Owing to equations (8 . 3 . 1) it follows that
Cj" 1 C a " ---- = M exp { - j* (i w : + v z > a + . . .)}
where M is equal to M/ 1 M. 2 V * . . . .
and Q is the heat of reaction per gram molecule of re-
actants or resultants. Thus the equilibrium constant
makes its appearance again as a quantity proportional to
exp ( Q/R0) leading to
dHogK^Q^
dd R0 2 '
CHAPTER IX
INTEBMOLECIJLAB FOBCES
9 . 1 The Effect of the Finite Sizes of Molecules. In
deriving the formula for the pressure of a gas in section (4 . 2)
it was implicitly assumed that we were dealing with a swarm
of point-particles. But the impossibility of crushing a body
of liquid or solid into an infinitesimal volume is simple and
direct evidence that whatever be the structure of molecules,
they have finite size in the sense that the centres of two
molecules cannot be forced nearer to one another than a
certain definite distance, minute though it be. Contrasting
two molecular systems, therefore, each one containing the
same number of molecules in the same volume, but one being
constituted of larger molecules than the other, it will be
seen that the mean free path between collisions will be a
trifle shorter in the first case than in the second. This will
have the effect of slightly increasing the rate of transfer of
molecules across an element of surface, thus producing at
the same temperature, i.e., at the same mean velocity, a
somewhat enhanced rate of transference of momentum.
The effect will be all the greater the larger the molecular
size in comparison with mean free path, i.e., the larger the
concentration. We may, therefore, infer on general grounds
that the pressure of a gas whose concentration is v molecules
per unit volume is given more accurately than before by
some formula, such as
p = v iff (v) kO,
where (v) is a function which approaches unity as v
approaches zero, and increases in value as v increases.
Expanding iff (v) as a series in ascending powers of v, we can
as a first approximation retain the first power of v only and
write
2> = v(l +0v)&0 . . . (9.1.1)
92
INTERMOLECULAR FORCES 93
as a somewhat amended form of the simple Boyle's law.
From the reasoning employed, it will be realised that as
between different gases, the constant /J will be larger for
larger molecules. We have in the reasoning implicitly
idealised a collision as an instantaneous phenomenon. It is
scarcely probable that the actual occurrence is dynamically
so simple ; still it is evident from the cohesion of solid and
liquid matter and the broad facts of their compressibilities,
that the repulsive forces called into play at the close en-
counter of two molecules disappears at a very small distance
apart, and so the conversion of kinetic energy into potential
energy during the encounter occupies a very brief time as
compared with the mean interval between collisions. It is
this fact concerning the intermolecular repulsive forces
exerted at very near approach which allows us to dispose of
them in the somewhat cavalier manner employed above.
9 . 2 Intermolecular Attraction. The elementary applica-
tion of the conception of molecular attraction to the explana-
tion of latent heat of vaporisation of a liquid is no doubt
known to the reader. In the interior of the fluid a molecule
does not experience a constant force in any definite direction,
as it is surrounded by molecules whose resultant pull on it
will be on the average zero. Only in the molecular layer at
the surface whose thickness is equal to the radius of mole-
cular attraction (beyond which the force becomes negligible)
will there be a resultant pull normally inwards. This can be
regarded as equivalent to removing the molecular attraction
and replacing it by an increased external pressure. The
amount of this increase is not difficult to estimate. Let/ (r)
represent the magnitude of the force between two molecules
separated by a distance r. Consider a particular molecule,
A, and the molecules surrounding it in a spherical shell
between spheres of radii r and r + 8r. Suppose the fluid to
expand uniformly by a small amount so that there is a linear
coefficient of extension of value e ; as the mutual potential
energy of two molecules will increase by the amount / (r) er,
it follows that the increase in the mutual potential energy
of A and its neighbours in the shell will increase by
rf(r) vlnr 2 Sr.
94 STATISTICAL MECHANICS FOR STUDENTS
Hence the increase in the mutual potential energy of A and
all its neighbours is given by the integral
4:776 V
|>/(r)dr
where or is the nearest distance of approach of molecular
centres. The increase in the whole potential energy of
attraction of the molecules will be obtained by summing this
for all) the molecules and taking half the sum. (Otherwise
the mutual energy of any pair of molecules would be counted
twice.) The result is
2 TT e n v
where n is the total number of molecules, so that v = n/v.
But if an increase of volume 8v accompanies the linear
extension , then e = | Sv/v, and so the result for the
increase in potential energy is
a v 2 8v,
where a = ~ ("r 3 f(r) dr . . . (9.2.1)
If the work thus done in the expansion had been performed
against an external pressure &>, instead of against molecular
attraction, its value would have been to o> Sv. Thus the
change of momentum produced in the molecules by the
cohesion exerted in the molecular layer is the same as that
produced by an external pressure of amount a i> 2 . This
correction shows that the internal pressure, i.e., the rate of
transference of normal momentum across unit area in the
interior of the fluid, is p + a v* and not p. So, recalling the
correction made in the previous section, we have
p -fa v 2 = v(l +pv)k9 . . (9.2.2)
If we write a = a tit and b j8 n, we obtain as a better
approximation to the equation of state than Boyle's law the
following result
. a nlcO /. . b
p +- =
INTERMOLECULAR FORCES 95
or putting (1 + bjv)~ l approximately equal to 1 bjv
v-b)=KO (9-2.3)
This is Van der Waal's famous equation, and of the various
interesting conclusions to be drawn from it the reader can
inform himself in text-books of physics or physical chemistry.
We are here concerned with the effect produced on our
statistical methods by the introduction of intermolecular
force into the arguments, and we shall therefore concentrate
on one result which follows from a study of the rela-
tion (9.2. 3).
It is well known that if isothermal curves are drawn,
using Van der Waal's equation, these curves show a charac-
teristic feature when the temperature is low enough. Travel-
ling along such an isothermal in the sense of increasing
volume, the pressure diminishes to a minimum, then in-
creases for a space, reaches a maximum value, and once more
proceeds to dimmish indefinitely. There are, in fact, two
values of v where dp/dv is zero, and between them dp/dv is
positive. Within this region of pressure there are, in fact,
three values of v mathematically possible for each value of
p (9, of course, being given).* The one which lies on the
part where dp/dv is positive is considered to be so physically
unstable as to escape observation by reason of its transience
if it were produced. Of the other two, one is considered to
correspond to a vapour phase which may be in an absolutely
stable unsaturated or saturated state, or in a less stable
condition of supersaturation ; the remaining value of v is
associated with the liquid state, which in its turn may be in
a relatively unstable superheated condition, or quite stable
below its boiling point at the pressure. This interpretation,
of course, implies that the temperature below which the
particular form of the isothermals manifests itself is the
critical temperature of the substance. In this way is the
* Since (9 . 2 . 3) is for given p and a cubic equation in i>, there are, of
course, three values of v possible for any value of p, whether within the
range specified or no. But, of course, two of these may be imaginary, or
if all are real, they will, if not in the range, lie on those parts of the curve
(which, be it noted, has really two branches, one not being usually shown
in the books) for which v < 6 with which we are not physically concerned.
96 STATISTICAL MECHANICS FOR STUDENTS
equation (9.2.3) linked up with the hypothesis that there
is a continuity between the liquid and gaseous states of
aggregation through intermediate homogeneous states which,
however, although physically conceivable, could only have a
very transient existence if actually produced . Before passing
on to consider this fact from the point of view of the statis-
tical methods employed in previous chapters, we may
realise its possibility in a general way, apart from special
analysis, as follows. The internal pressure, R0/(v 6), is
reduced by the cohesion, a/v 2 , in the surface-layer to the
value, p, of the pressure actually observed. An increase in
volume implies a diminished density, and therefore a decrease
in the internal pressure, i.e., a smaller rush of molecules
across the inner surface of the molecular layer. But the
diminished density also produces a diminished cohesion,
and therefore less hindrance to these molecules in escaping
across the outer surface of the layer, so that the net result
might be actually a greater external pressure. Students,
through a failure to grasp the real meaning of the symbol p,
sometimes fall into the error of imagining that a homo-
geneous state of aggregation in which p increases with v,
is physically impossible, " because,' 5 as they say, " it is
impossible for the pressure to increase if the volume in-
creases," thereby betraying the fact that they think that
the symbol^ refers to the internal pressure, concerning which
the statement is true enough. The very great instability of
such a state (not its impossibility in an absolute sense) can,
however, be realised by conceiving it to exist and then
considering what would happen if by reason of the mole-
cular motion a small fluctuation began in the density of a
small portion of it. In the ordinary way, if a small portion
of a fluid expands, there is a reduced outrush of molecules
from its original volume. The surrounding fluid is pouring
in molecules at the normal rate, with the result that further
fluctuation in this direction is checked, and a similar con-
clusion follows for a fluctuation involving transient increase
of density. But in the state of aggregation imagined, this
would not take place ; the outrush is accompanied by a
reduction in cohesion of such magnitude that the fluctuation
INTERMOLECULAR FORCES 97
is not checked, but actually assisted to greater intensity, as
the increasing outrush overwhelms the normal stream of
molecules inwards. Thus would be set up an expansion of
the original fluctuating element to a less dense and more
stable state. Similarly an original fluctuation in some ele-
ment, beginning with an increase of density, instead of being
checked by an enhanced outflow of molecules from it, would
be assisted on account of a too great increase in cohesion,
and a consequent inhibition of the balancing outward stream,
so that the normal inward stream from without would force
molecules into the element and so produce a stable con-
densed phase therein. In some such way we can visualise
the separation of the unstable phase into droplets of liquid
and a saturated vapour phase.
9 . 3 The Probability of a Macroscopic State when Inter-
molecular Action is Involved. The change introduced into
the analysis of Chapter III. by the assumption of inter-mole-
cular force is produced by the fact that the energy of a state
specified by the numbers n l9 7i 2 , ...... , n c of representative
points in the phase -cells is no longer a linear function of those
numbers, as was the case when we wrote E equal to 2 r n r ,
the r being functions of the parameters of the system. If
we write a general function / (n l9 n%, ...... n c , a v a 2 ,
...... aj, or briefly/ (n, a) for the energy of the state, the
variational equation (3.2. 10) in section (3 . 2) must be
replaced by
leading to the result that we must determine the values of
the n r in the most probable state as well as the functions A, fj,
in terms of n, E, and the parameters by means of the c + 2
equations
2 n r = n
f (n, a) = E
As regards the function, / (n, a), it will, apart from the
linear terms in n r (which involve kinetic energy and potential
S.M. H
98 STATISTICAL MECHANICS FOR STUDENTS
energy arising from external bodies), depend on the quantities
Z (n r ) 8 , this summation referring to the sum of the numbers
n r over those phase-cells which have one space-element in
common. If the c phase-cells are constituted by associating
each one of a elements of volume with each one of /? elements
of extension-in-momentum (c being therefore equal to a /?),
then this summation is over the j8 phase-cells which have
the 5 th element of volume in common. Calling the sum N,,
we have
c /
/ (n, a) = S r n r + </r (N^ N 2 , .... N a a l9 a 2 . . . . aj,
r-l
r being the sum of the kinetic and external potential
energies in the r th phase-cell. Hence
df^a) ^ (N, a) 3N,
dn r r 3N, dn r
a (N, a)
~~ 9N. "
= *, +Xs (Ni> N 2 , NJ.
Here x (N 1? N 2 , N tt ) can be regarded as the increase
in mutual potential energy produced by introducing one more
molecule into the element of volume which is a constituent
part of the r ih phase-cell. If we assume that the phase-cells
are constructed in such manner that their constituent volume
elements are considerably larger than the sphere of mole-
cular interaction (which from what we know of the range of
molecular forces is not at variance with the assumption of
the physical smallness of those elements), we can take
^, (Nj, N 2 , NJ to be the mutual potential energy of
one molecule with respect to all the rest in the s th volume-
element. It will, therefore, be a function of the concentration
of the molecules in this element which we shall denote by v t ;
so we write, instead of ^ (N x , N 2 , NJ, the functional
form <f> (v g ) of the one variable v 8 . If we could assume some
simple law of force for the attractions, and if we could assume
this law to hold down to any distance apart (virtually
assuming the molecules to be point-centres of force), <f> (vj
would have the form a v^ a being a positive constant, and
INTERMOLECULAR FORCES 99
the minus sign being due to the fact that for attractive
forces potential energy decreases with decreasing separation
of molecules and increasing concentration. But, of course,
this overlooks the occurrence of intermolecular repulsions
at close encounter. The fact that there is a finite limit to
the value of v under the greatest pressures conceivable,
requires us to assume that although <f> (v) may very well
behave as a v for relatively small values of v, yet for
values approaching some maximum limit v , <{> (v) must
begin to increase in value not only up to zero once more, but
actually to positive infinity if we are to regard concentrations
beyond v as physically impossible. This is in fact the
counterpart of the infinite value required for the pressure p
in Van der Waal's equation to reduce v to the value h.
From these equations we now find for the number of
molecules with representative points in the r th phase-cell,
in the most probable state the value
This, by summation over the /? elements of extension-in-
momentum, which linked up with the s ih element of volume
yield /? phase-cells, gives N, as
ft
C i*.<b(v K ) y 0ntf /Q o i \
6 ' Z/ 6 ~ , . , . ^y . O . 1 j
N 8 , if divided by the magnitude of the element of volume,
gives v g , and since the summation can be as usual replaced
by an integration over all values of momentum, we obtain
finally the result
+ 00 -f QO + 00
Vg == D e-^W J J J e ""^ d dr ) d $
** (V J . . . (9.3.2)
where D is a constant. We are disregarding any external
field of force.
From (9 . 3 . 1) it follows just as before that 3/(2 p) is the
average kinetic energy of a molecule, in any element of
volume, for the factor e"" M * ( " ) , being the same for all phase-
H 2
100 STATISTICAL MECHANICS FOR STUDENTS
cells associated with that element of volume, does not invali-
date any of the steps occurring in the calculation. Thus
internal potential energy does not interfere any more than
external with the validity of the law of energy-partition.
The result (9.3.2) shows that the most probable state is
consistent with a uniform spatial density given by any root
of the equation in
v = A /J e~^ (v) , . . . (9.3.3)
where A is a constant.
The easiest way to see how many roots are involved is to
plot two graphs
y = e -^ ( *> (9.3.5)
the co-ordinate x representing the concentration v. The roots
of (9 . 3 . 3) are the abscissae of the points of intersection of the
graphs (9.3.4) and (9.3. 5). Now if, as suggested by the
neglect of forces of repulsion, <f> (x) were taken to be a x,
there would either be two roots or none, according as the
straight line y = &/A/J, cut or did not cut the exponential
curve, y = e* ax . But if we take account of the finite size
of the molecules, <j> (x), although increasing in proportion
to x at first, will, as x gets very near to v in value, pass
through a maximum and decrease through zero to minus
infinity at x = v . Hence the curve, y e~^ (x \ will at a
certain point in its exponential ascent abruptly turn down
and y will decrease to zero at x = v .
The curve y = e ""
INTERMOLECULAR FORCES 101
We thus see the possibility of the straight line (9.3.4)
cutting the curve (9 . 3 . 5) in three points. If, however, //, is
reduced in value, i.e., the temperature raised, the slope of
the line will be increased, and the peak of the curve will not
be so high, and so the line would only cut the curve in one
point corresponding to a relatively small value of v. If on
the other hand /x is increased, the line will fall low, the curve
will reach high ; once more there will be only one point of
intersection, this time, however, yielding a concentration
close to v . We thus reproduce once more the result usually
derived from Van der Waal's equation. The author has not
met elsewhere this method of presenting the matter, but
thinks it is perhaps worth while to outline it here, as it links
up the result with the usual statistical method of estimating
probabilities by counting complexions consistent with the
energy conditions. However, in order to bring out the great
instability of the state indicated by the point Q, we must
appeal to a very ingenious piece of analysis first given by
Smoluchowski in 1904. It arises in connection with the
mathematical discussion of fluctuations in density of a
molecular system. This will be dealt with in the next
chapter.
CHAPTER X
FLUCTUATIONS OF DENSITY IN A MOLECULAR SYSTEM
10.1 The Probability o! the Occurrence of a Prescribed
Number of Molecules in an Element of Volume. Reverting
to the fundamental formula of the theory, we find that when
uniform concentration of molecules in a definite volume is
the most probable state (there being no external field of
force), the number of complexions associated with a state in
which n l9 n 2 , ...... , n c molecules occur in the c elements
of volume respectively, is
_ n\ _
njnj. ...".. .n e \ . . . . (10 . 1 . 1)
Hence the probability that there shall be a definite num-
ber, Z, molecules in the r th volume element is proportional to
the sum of expressions such as (10 . 1 . 1) for all the states in
which n r I, i.e., to
1 27 _ U - _ (10.1.2)
l\ fti f . n 2 \ ...... n,^ n r ^\ ..... n c \
the summation being for all positive integral values of
H!, n 2 , ...... ra r _i, w rfl , ...... n c consistent with
If (n I) \ were substituted for n ! in the numerator, the
terms in this summation would be the coefficients of the
x a y ft z* ...... products in the expansion of
(x + y + z + ...... toe 1 terms)""'
Hence the expression (10 . 1 . 2) is equal to
(n - I) \ I ! v
But c is a large number, and the last factor is practically
c w /c*. Also, as we know that large deviations of I from the
102
FLUCTUATIONS OF DENSITY 103
average value n/c are rare, I is very much smaller than n
and we can write n 1 for n (n 1) (n 2) (n I + 1).
Thus (since c n is a constant), we find that the probability
sought is proportional to
n 1 a 1
where a is the average value of the molecules in an element
of volume. Finally, since the sum of these probabilities for
all values of I is to be unity, and since the sum of the expres-
sions a l /l ! for all values of I is e a , we see that the probability
that there shall be / molecules in an element of volume, the
average number being a, turns out to be
e~* ..... (10.1.3)
A simple scrutiny of the result shows that a l /l ! increases
as / increases up to a, and thereafter decreases as I travels
beyond a in value ; thus there is a maximum probability
for / = a, as there should be, of course. The average value
of the deviation I a is, of course, zero, but the average
value of its numerical magnitude is not. It is, as usual,
estimated by means of the average value of the square of
I a. This will be
_ a | la 1
' s
By a rearrangement of terms, this can be written
,- r(
104 STATISTICAL MECHANICS FOR STUDENTS
Thus the mean deviation from the average value a of
the number in the element is a*. Clearly this mean
deviation becomes proportionally less as the concentration
in the element increases. For a million molecules in an
element on the average, the mean deviation is a thousand
either way, or one in a thousand. For a hundred million
molecules in the element, the deviation is one in ten thousand.
This proportional deviation measured thus, is called the
" condensation/' and it is clear that the average condensation
in an element holding a molecules on the average is 1 in a*
or a~*.
Returning to (10 . 1 . 3), we see that the ratio of this expres-
sion to its maximum value is
a 1 la a
/
/!/!
a l -*a\
or
l\
Taking I as greater than a, this can be written
a. a. a.
(a + 1) (a + 2) ...... I'
there being j factors in numerator or denominator where j
is the deviation I a. Hence the logarithm of this ratio is
which = -- ...... - - + negligible terms
2a
where, as usual, we disregard as of no importance the states
where j would become more than a small fraction of a. If
I < a and j = a I, the ratio is
a (a -r 1) I
a. . a a
and the same result emerges for the logarithm of this ratio.
If we now wish to convert this result into a form suitable
for dealing with continuous changes of density within a
volume element rather than discrete changes in numerical
FLUCTUATIONS OF DENSITY 105
concentration, we replace j (j + l)/2a by ^ a y 2 , where y is
the condensation in density, j/a ; for in this case unity will
be negligible compared with the values of j since the average
value of j is a* and is actually an enormous number even
in a physically small element. Thus if the density p is
given by
P=fc,(l+y),
where p is the average density, the probability that the
density in the element lies between p and p -f- Sp, i.e., the
condensation y between y and y -f- Sy is P(y) Sy, P(y) being
a function which satisfies
where P is the maximum value of P(y), occurring when
y = 0. P is readily obtained from the fact that
P(y) dy = I*
-CO
and by a reference to the Appendix to Chapter I., turns out
to be (a/2 TT)*. Thus the probability that the condensation
is in the range y to y + Sy is
/ay
\2W
e * Sy . . . . (10.1.4)
a being the average number of molecules in the element.
The chance of a particular condensation is smaller the greater
the value of a ; this we have already deduced, but it is once
more evident from (10 . 1 . 4), the decreasing exponential
factor quite easily swamping the increasing factor a*.
From (10 . 1 . 4) we can once more calculate the average value
of the condensation, i.e., the root-mean-square condensation.
It is
a \* f+
J k
* The reader may think it absurd to integrate from negative infinity
for 7, since by definition the numerically greatest value of 7, i.e. (I a)/a,
on its negative side is unity when I ia zero. Fortunately we are rescued
from this apparent absurdity by the fact that by the time ay 2 / 2 has reached
a numerical value such as 10, the outlying parts of the integral are negli-
gible, and on account of the great value of a this is attained by quite small
values of 7 positive or negative.
106 STATISTICAL MECHANICS FOR STUDENTS
, . , / a \* 1 TT*
which = I ) . - |
\2W 2 (a/2)*
_ 1
>
a
so that the root-mean-square of y is a~* as determined earlier.
Calling the mean square value y 2 , we can write (10 . 1 . 4) in
the form
^ exp { %= )Sy.
27r/ V 2yV
10 . 2 Smoluchowski's Theory of the Unstable States of a
Fluid. The usual presentations of Smoluchowski's theory
are so brief in the initial statements leading the funda-
mental equation of the theory that there is an element of
obscurity and vagueness in the mind of the beginner con-
cerning its validity. It is hoped that the following preamble
will remove such doubt.
The fluctuations of density which occur throughout the
body of a fluid owing to the molecular motion, imply that
the volume of a given number of molecules is always varying.
Instead, therefore, of considering the fluid as a molecular
system, let us for the moment think of it as a continuous
medium at rest in the broad sense, but pulsating throughout
with compressions and rarefactions. Let us further conceive
it to be divided into elements of volume, containing the same
mass, which will therefore be equal if the density be truly
uniform throughout. Let there be N of these elements, and
for the moment let us consider that each of them is capable
of having by reason of the pulsations in size any one of c
discrete values of volume, v lt v 2 , v c . (We are here,
just as in Chapters I. to III., compelled to adopt in the
initial stages of the reasoning the standpoint of discon-
tinuity which will be later modified to suit the requirements
of continuity.) The reader will now easily realise that a
complexion of this system will be specified by saying that
Nj particular elements have a volume v l9 N 2 particular
elements a volume v 2 , etc. The number of complexions is
as usual N !/Nj ! N 2 ! N c ! for the statistical state
FLUCTUATIONS OF DENSITY 107
Nj, N 2 , , N c . As usual, this will be assumed to be pro-
portional to the probability of the occurrence of the state.
To find the most probable state, we proceed as before,
taking account of the energy condition. This condition is
obtained in the following manner. If v is the volume of any
element at uniform density throughout, and p the uniform
pressure then existing, the potential energy in an element
of the fluid at volume v r over and above the energy at
volume v n is
(10.2.1)
p being the pressure at volume v. To justify this, remember
that if v is smaller than v 09 p is greater than p oy and so work
would have to be done on the element to force it into a
smaller volume than that consistent with a pressure p from
its environment. (The case is analogous to that of a body
suspended at the end of a spring. If the tension T at a
length I is greater than the weight w, then work has to be
done in forcing the body down to this length of the spring.)
This work is the product of p p , and the decrease in
volume, i.e., the product of p p and $v. If, on the other
hand, v is larger than v and p less than p ot work still has to
be done on the element to force it out into a larger volume
than that consistent with the pressure p . (Again, there is
the analogy of the body being higher than its mean position,
there being a reduced energy of extension of the spring, but
a more than compensating gain in the gravitational energy
of the body.) The work done is the product of p p, and
the increase in volume, i.e., (p p) Sv. As in the case of a
body oscillating up and down there is increase of potential
energy whether the body is above or below its mean position,
so in the pulsating element of volume there is increased
potential energy whether the size of the element is greater
or less than v . The reader will naturally think of the
accompanying changes of kinetic energy which would accom-
pany such oscillations and pulsations in isolated systems ;
but he is asked to bear in mind that we are counting com-
plexions consistent with constant temperature throughout, i.e.,
108 STATISTICAL MECHANICS FOR STUDENTS
changes in kinetic energy are ruled out by that proviso, and
the energy condition to be satisfied is that gains in the
potential energy of certain elements are to be compensated
by losses in the others, or, in short, if we represent (10 . 2 . 1)
by r
N! X + N 2 e 2 + ...... + N c c = constant.
Combined with the condition of a constant sum for the
N f , we have the same solution as before. In the most
probable state of the pulsating medium the number of
elements whose volume will be v r at any instant is pro-
portional to exp ( r] e r ), where 77 is a constant to be
determined from the condition of constant number and
potential energy.
The equation of state of the fluid enables us to express
(10 . 2 . 1) as a function of v r . (It may be as well to point out
at this juncture, in view of the warning issued some pages
back, that in (10 . 2 . 1) p is the external pressure ; the calcu-
lation of work done on and by the element in its changes
ensures that.) If a, 6, c, etc., stand for the partial differential
coefficients dp/d v, d*p/dv 2 , d 3 p/dv 3 , etc., all estimated at the
value v for t;, and, of course, at constant temperature, then
we know by Taylor's theorem that
P =Po +~j (V - V ) + - (V - V )* + - (0 -. t?,) s + ......
and thus by (10.2. 1)
We can now adapt the preceding considerations to the
idea that the volume of an element changes continuously,
and not by discrete amounts. The number of elements which
in the most probable state have a volume between v and
v + 8v is given by
D exp fa iff
where
o 1
FLUCTUATIONS OF DENSITY 109
and D and rj are constants to be determined from
o
* dv = N
D !
Jo
D
* a
e n *dv = constant.
Instead of the variable v, we can introduce the condensa-
tion, y, defined as (v - v)/v . The number of elements
which in the most probable state have a value of condensa-
tion between y and y + Sy is proportional to
where
</> (y) = Ay 2 + By 3 + Cy 4 + ....... (10 . 2 . 2)
A, B, C, ...... being coefficients given by
A-
2!
< 10 - 2 - 3 >
If now we suppose that our attention is concentrated on
one element of the fluid, we can assume the probability
that at a given moment the condensation in this element has
a value between y' and y" to be proportional to the number
of elements which at any time have in the most probable
state condensations within these limits. This probability is
therefore proportional to
J e&p
We can find the value of 77 directly by a reference to the
result (10 . 1 . 4) in the previous section. In the case of gases
in a rare state for which one would anticipate that the
analysis of that section holds, the molecules exerting no
action on one another on the average, it appears that 77 <f> (y)
110 STATISTICAL MECHANICS FOR STUDENTS
should be equal to ay 2 /2, where a is the average number of
molecules in the element of volume. Also as only relatively
small values of y are effective in actual fact, we can ignore y 3
and higher powers of y in the series for <(y) in the case
contemplated. Hence
77 A y 2 = - a y 2 /2
02
or T? - -- / <
2 ! \3v
_ a I ( ajc9 \
V/ \ T?"/
ak9
P =
J_^
Once more the distribution constant emerges and Smolu-
chowski's result is written
carp/ft <ji(y)Uy . . .. (10.2.4)
The author trusts that the rather lengthy exposition is
justified. Most accounts start baldly with the assumption
that the probability in question is given by an expression
involving
-JL
ke
dv,
r
where E = I (p p ) dv, with no very clear reason given
Jv
for this value.*
In order to apply the result to other states of the fluid
than that of rare gas, we first of all note that if (dp/dv) is
negative, the index in the exponential factor of the prob-
ability is essentially negative for any value of y positive
or negative, giving decreasing probability for increasing
fluctuations, obviously a necessity of any stable state. If,
* It will be perhaps wise" to warn the reader that E is more usually
written
f'
a (p
Jv
po)dv,
but that is because v is then the specific volume of the fluid, and so the
factor, ma, the mass of the fluid in an element of volume is required. In
the text v is the volume of the element.
FLUCTUATIONS OF DENSITY 111
however, (dp/dv) is positive in the state of uniform distri-
bution and we have seen the theoretical possibility of this
the exponent is positive for any value of y, giving increasing
probability for increasing condensation or rarefaction ; in
short, the state is highly unstable, and so the general in-
ference drawn before is amply confirmed by this closer
analysis. It will be remembered that on a Van der Waal's
isotherm there may be two points where dpjdv is zero. If
the average density of a fluid should correspond to either of
these points, the series for c(y) would start with y 3 , and
ignoring the remaining terms of the series, the probability
would involve
Thus, according to the sign of d 2 p/dv 2 , we would have
increasing probability for fluctuations involving expansion,
and decreasing probability for fluctuations involving con-
traction or vice versd. This one-sided stability would be
useless for maintaining a physically homogeneous state.
Indeed, there are parts of the isothermals where dp/dv is
negative, but which are so near to the points just mentioned
that they can hardly be said to correspond to stability in a
real physical sense, since they are associated with super-
saturated vapour or superheated liquid. Smoluchowski's
result is quite consistent with this ; for a negative but small
value of A in e^ AY ' means that y must increase numerically
more than usual before one goes beyond ordinary prob-
abilities, and by that time the fluid in an element may, if
the fluctuation has been in the suitable sense, have attained
the highly unstable condition.
But perhaps the most interesting application of Smolu-
chowski's theory concerns the critical state at which dp/dv
and d 2 p/dv 2 are both zero. In that case both A and B are
zero, and the series for <f>(y) begins with Cy 4 . To determine
the stability of this state we must determine a value for C.
This can at all events be obtained approximately from Van
der Waal's equation, which we write
___ akO a
P ~~ v 3 ~~v*'
112 STATISTICAL MECHANICS FOR STUDENTS
(We are using a for the cohesion constant, since a is being
used as the average number of molecules in an element of
volume, and for the minimum volume.)
Thus
24a
W ~~ ( V - 0)4
and by (10. 2. 3)
v 4 (v 0) 4
At the critical point it is known (since dp/dv and d 2 p/dv 2
are zero there) that v = 30, p = a/270 2 and ak0 = 8a/270.
Thus at that point
a Slakd
Slake
8 64
9a
The series for <(y) now starts with y 4 , and neglecting higher
powers, the probability now involves the exponential factor
exp
As this decreases with any variation of y from zero, the
critical state is stable ; but closer investigation shows that
the condensation fluctuates between wider values than in
other conditions. This is proved by working out an average
value for y. Thus the average value of the positive con-
densations is (writing c for 9a/64)
which, by writing x for c*y, becomes
JOO
e~ x * x dx
{00
e~* 4 dx
FLUCTUATIONS OF DENSITY 113
1 f
The integral in the numerator is I eT^ dy, which is,
2 Jo
as we know, 7r*/4 ; the integral in the denominator can be
calculated by quadrature to be 0-666 . . ., and so we obtain
finally for the average value of the positive condensation
1.13
It is the appearance of the fourth root of a, and not the
square root, as in the case of gases in a more usual con-
dition, which is the interesting feature. Thus, in an element
of volume containing 10 8 molecules, which for a substance
in the critical state would have linear dimensions of the
order of magnitude of the wavelengths of light, the value
of this expression would be about *01, and so fluctuations in
size of the order of 1 per cent, in density would be the
average sort of occurrence. It is this result which is held to
account for the well-known opalcscence which appears in a
substance in the critical condition ; the real lack of homo-
geneity in the medium is sufficiently marked to scatter the
light of a beam which is passed through it. When the
illumination is strong, a scattering of bluish light can be
observed in a direction at right angles to the beam. Keesom
has, in fact, linked up Smoluchowski's theory with physical
optics, and deduced the well-known formula of Rayleigh
for the intensity of the blue light of the sky, thus con-
necting this phenomenon with fluctuation theory.
CHAPTER XI
THE SECOND LAW OF THERMODYNAMICS . II
11.1 The Thermodynamical Equilibrium of a Condensed
System. In Chapter VI. we deduced the second law of
thermodynamics for a gaseous phase in statistical equi-
librium. It is necessary to make sure that the deduction is
still valid for systems in which we cannot ignore inter-
molecular action.
As we have seen in Chapter IX., the normal state of a
system, for which the energy is given by a general function
/ (n l9 n 2 , n c , a lt a 2 , a c ) of the numbers in the
phase-cells and the parameters, is obtained from the equa-
tions
/ (n l9 n 2 , n c ,a l9 a 2 , a e ) = E
% + ^ 2 + +n c = n . . (11.1.1)
3f(n 9 a)
and the c equations, log n r = A //,-
dn r
These may be regarded as c -f- 2 equations to determine
A, E and each of the n r as functions of ft and the e parameters
a l9 a> 2 > ...... 9 a e- We shall denote the functions so deter-
mined by A (/A, a), H(fi, a), i/^fi, a), ...... , v c (p,, a). H(ft,a)
is, of course, obtained from/(n, a) by inserting the c functions
v r (ft, a) in place of the variables, n r ; i.e.,
H(ft,a) =/K,v 2 , ...... " c ,ai, a 2 > ...... > a e) (H.1.2)
Let us also denote by x r (/^> a ) ^he function obtained when
the functions ^ 1 (ft, a), *> 2 (ft, a), etc., are substituted for n l9 n 2 ,
etc., in df(n, a)/dn r ~ and by a (ft, a), the function obtained
when the same substitution is made in df(n y a)/da s .
It follows that
dv,
114
SECOND LAW OF THERMODYNAMICS 115
, 9H (u,(i) dv r , y / x /i-i i ^\
and ^ ' = Z x r i~- + O, ) - (H . 1 . 4)
ca 8 r ~i ca s
Recalling the general line of argument in Chapter VI., we
consider a change from a normal state with values /*, a for
the S.M. variables to a normal state with values fji + S//,,
a + Sa. The change of energy 8E which is given by
8a, . (11.1.5)
ca
is, as before, separated into two parts, one of which, SEj,
is the change accompanying a variation in the parameters,
but with the distribution in the phase-cells still left at the
original normal distribution. The remainder is 8E 2 . It
should be noted that SEj, the analogue of the mechanical
work alone done on the system, is not given by the second
term on the right-hand side of (11.1.5), as a glance at
(11.1.4) will show. Actually
, a) Sa,
^_' S Zx r ^Sa. . (11.1.6)
r .
=i ca 8 r-i-i ca 8
To proceed successfully from this point, our aim, as in
Chapter VI., must be to discover a function ^(/^ a), such that
8El = Z l Sa, . . . (11.1.7)
=i da g
and with that end in view, we write (11 . 1 . 6) as
8E X = * j H^ a) - 2 X, v r \ Sa, + Z Z v, ^ Sa,
8~ica 8 ( r =i ) r=i*-i va B
(11.1.8)
and endeavour to adapt the last term in this to the necessary
form. Guided by the procedure in Chapter VI., we obtain
from (11.1. 1) the following c equations, which are identically
true for any values of the a e
log v r (iJL, a) = A(^, a) p x r (fji y a) . (11.1.9)
12
,116 STATISTICAL MECHANICS FOR STUDENTS
Differentiating this with respect to any of the a s yields an
identity
9*> 9A 9v,
_ : = v r -- jLt V r
da, da, da t
and an addition of these gives
9x, 9A dn
a Z v r -^ = n since - = 0.
r ~i ca s da, ca 8
On substituting in (1 1 . 1 . 8), we find that (1 1 . 1 . 7) is valid
if we define M* 1 (/x, a) thus
This is obviously a generalisation of the definition in
Chapter VI. ; for if f(n, a) is linear in the n r , f(n, a) =
H n r df/dn r) and thus H (/x, a) = 27 x r ^ r ; so that in such
case "^ reduces to n A//*.
However, we have not carried the attempt to a successful
conclusion yet. We must also see if the mathematical
relation (6 . 2 . 7) in Chapter VI. is still valid. Naturally we
follow the same procedure as we did there and differentiate
the identities (11.1.9) with respect to p, thus obtaining
^v---vx- v
Sfji dfji
Addition of these c relations yields
nf-=Z v r Xr +M Zv*
Cfl r ~l r-1 C7jLt
From (11.1.10) we obtain
, a) A 9 Xr rv 1v ndX ^A
^/ v r ^j \ r -f- - - n ~.
- - ----- - - r r -
Cp, CfJL r=l OfJl. r=-l Of* (lOfJ. H
which by (11.1. 3) and (11 . 1 . 11)
" 1 A A
= - 2 v, X f - - a ,
/A r=l i
and by (11.1.10)
^ _ .(11.1.12)
SECOND LAW OF THERMODYNAMICS 117
Thus, as before,
and
M SE 2 = 8[/i{(H/i,a) -(/*, a)} ].
The general line of argument proceeds as before to a
deduction of the second law with the entropy defined by
4>(/x,o)=fc/i{H(^o) -(,*, a)},
for the relations (11.1.9) give us
c
log W w = ^ log n E v f log v r
r=l
c
= 7i log n n X -\- fji S v r x r ,
r=l
so that
* (log W m - n log tt) = fyijHGu, a)- ^(/x, a)}
and the connection between increasing entropy and increas-
ing probability is once more established.
CHAPTER XII
THE STATISTICAL-MECHANICAL THEORY OF A LIQUID AND A
VAPOUR PHASE IN CONTACT
12.1 Deduction of Clapeyron's Equation. An enclosure of
given volume is supposed to have a portion of its volume V l
occupied by a saturated vapour and the remainder V 2
occupied by the liquid ; the number of molecules in the
vapour is n l and n 2 in the liquid, so that the concentrations
are given by v x = n l /V l and i> 2 = n 2 /V 2 . In Chapter IX.
it was demonstrated that under these circumstances
-^i) . . . (12.1.1)
v '
\27rm
v 2 = T> (-V
\277-ra
where </>(v) is the potential energy of one molecule due to
intermolecular force in a place where the concentration is v.
This result is not dependent on any particular functional
form for </> ; but it will be found as we proceed that we must
have a modicum of information about it in order to come to
a definite conclusion in the problem on which we are engaged.
First of all we know that <j)(v) approaches a maximum value
as v decreases, and beyond a certain concentration it is
practically constant ; so that in the gaseous phase we shall
assume that <fr'(v) is zero, where <j>'(v) is written for d<f>(v)/dv.
As v increases and the molecules approach one another on
the average, the potential diminishes so long as the forces
are attractive. But when the liquid state is reached, the
average separation is such that there is a compensation
between attractions and repulsions on the average, and
further compression would involve an increase of the
potential energy due to preponderance of repulsive force.
So that the concentrations in the liquid state would be such
118
LIQUID AND VAPOUR 119
that for any value of v in the narrow range involved, (f>(v)
would practically be a minimum, so that <f>'(v) would be
practically zero in this case also. Thus in order to give the
necessary definiteness to our problem, we shall assume that
the function < satisfies for the two phases in question
f(*i) = f(" 2 )=0 . . . (12.1.2)
By reason of (12 .1.1.)
Vl e^h) = ^ e ^(v,)
or
log ft! +'/^K) log V a = log n 2 + jot^(v a ) log V 2 .
Now consider the system with a slightly altered distri-
bution constant, i.e., temperature ; this will involve an
alteration in the numbers and concentrations as well as in
the volumes V t and V 2 . Differentiating with respect to the
temperature 6, and remembering that dfijdO = 1/&0 2 , we
obtain
__ . >
! dd V l dd "*" w {Vl) dd
^ _ _ _
i ~~n 2 d9 V 2 dd ' 2 d9
so that, on account of the hypothesis we make as regards an
ideal liquid and vapour state in (12. 1. 2),
^.
n! dd n 2 dO V l dd V 2 dO
(12.1.3)
We can simplify this result very markedly by assuming
that the two phases occupy equal parts of the enclosure at
the temperature 9. Since the enclosure has a fixed volume
dVJd9 = dV 2 /d9, and if we now also assume that Vj is
equal to V 2 , the two terms involving these quantities dis-
appear. Further under such circumstances
ni^v*
n 2 v l
where v l and v 2 are the specific volumes of the two phases,
since n v l m and n 2 v 2 m are equal respectively to V l and V 2 .
120 STATISTICAL MECHANICS FOR STUDENTS
Lastly, since the total number of molecules is constant,
dnjdd = - dn 2 jd0 9 and so the left-hand side of (12 . 1 . 3)
becomes
i l nj dO '
which
( 1 v 2 \ I dn l
"TjH^W
v z \ d log K!
On the right-hand side of (12. 1. 3), <t>(v^) <f>(v 2 ) is the
energy required to remove one molecule from the liquid
phase to the vapour, and since the volume of the enclosure
is fixed, and therefore no external work performed, this is
the internal heat of vaporisation per molecule. Denoting it
by w, we have, as the final result
... (12.1.4)
and this leads directly to Clapeyron's equation, for if p is the
vapour pressure,
p = j/i k6
and dlogp _d log Vl 1
~~dO dT~ +
v * w i ^
""
where L t - is the internal latent heat of n molecules and R =
nk. Since the first phase is considered to be an ideal gas,
R0 == pv ly and so
dlogp __ Vi LJ + p(v l v 2 )
_ l
==
LIQUID AND VAPOUR 121
where L is the ordinary latent heat. As v 2 is small compared
to v v we have approximately
d log p __ L
~ ~
but keeping the exact equation, and once more writing
for R#, we obtain
d log p _ L
d9 p0(v L v 2 )
d ?=- _ . . . (12.1.5)
de 6(v l - v z ) v '
which is the proper result, and as a matter of fact, is
the exact equation deducible by strictly thermodynamical
reasoning for real liquids and vapours.
12.2 The Relation between the Latent Heat and the Specific
Heats. The second thermodynamic equation for a liquid-
vapour system can now be easily obtained. Let s l and s 2
be the specific heats of the vapour and the liquid, it being
understood that these are the thermal capacities of unit
mass of the vapour or liquid as heated in the fixed enclosure
(not subjected to a constant pressure nor on the other hand
with each portion maintained at constant volume, the usual
conditions). The proviso is well known to those acquainted
with the thermodynamical treatment of this matter. The
vapour, for instance, is maintained in a saturated condition
during the heating, involving a diminution of volume, and
on that account it may happen that if the external work
thus performed on the vapour is too great, heat would have
to be removed from the vapour, and s l would be negative.
This is in fact the case with water. In the case of the liquid,
s 2 differs but little from the usual specific heat under constant
external pressure.
If now unit mass of the liquid is heated through 86, the
heat supplied is s 2 86 y and external work p8v 2 is performed.
Thus the internal energy of the unit mass of liquid is greater
by s 2 86 p8v 2 . If unit mass of the vapour is similarly
heated, its energy increases by s^O - pdv v (As just
mentioned, 8v l is in general a negative quantity if 86 is
122 STATISTICAL MECHANICS FOR STUDENTS
positive.) Hence the energy difference between the liquid
and vapour phase increases during the change of temperature
by
But this energy difference is L^ at first, and after the tem-
perature rise L; -f 8L, t , so
dd d0
Hence
d{L, + P(VI v 2 )\ .
de - - = s *~ s * +
and thus, by Clapeyron's equation
To be sure by our hypothesis (12.1.2)
dw
__ A
and L f will not change with temperature at all for our ideal
liquid, which is a rather restrictive hypothesis.
CHAPTER XIII
THE SOLID STATE CONSIDERED AS A SIMPLE LATTICE OB 1
MASSIVE PARTICLES
13 . 1 The Specific Heat of a Monatomic Solid. We now
come to the treatment on classical lines of the last topic to
be dealt with before an endeavour is made to introduce the
reader to the modifications of statistical-mechanical theory
occasioned by the quantum hypothesis. On classical lines
it lends itself to very simple treatment indeed.
The rigidity of a solid, which is its characteristic feature,
we shall idealise by conceiving it as constituted of a group
of n particles, each of mass m, situated at the points of a
simple space-lattice, so that choosing three axes of reference
in a suitable manner, the co-ordinates of any particle are
given by ja, kb.lc, where a, 6, c are three elementary lengths
and J, k, I are integers at least, that is assumed to be the
state of affairs at absolute zero of temperature with no
thermal motion going on, each particle being held firmly to
its equilibrium position by the forces arising from its neigh-
bours. The particles are regarded as without structure ;
so the model is " monatomic." If energy is given to the
system, each particle will vibrate about its mean position,
the displacements at any moment parallel to the axes being
represented by , 77, , and the velocity-components by
, 77, . If we make the well-known simple assumption that
the elastic forces on a particle set up by the displacements
are towards the equilibrium position and proportional to the
displacement, the motion of each particle is simple harmonic.
The energy is a quadratic function of the 3n co-ordinates
(., 7] r , r ), and the 3n momenta (m ., mr} r , m r ), involving
squares alone. The application of the statistical method
follows the usual course, with a phase -diagram, in which are
represented these 3n co-ordinates and 3n momenta, par-
123
124 STATISTICAL MECHANICS FOR STUDENTS
titioned into phase-cells. Counting of complexions, deter-
mination of the most probable state, etc., lead to the usual
type of solution as to the number of representative points
in a phase-cell, this number depending as ever on e~ Me
where e is the energy corresponding to the centre of the
phase-cell, and is, of course, the sum of kinetic and potential
parts. The distribution constant /*, is still connected with
the temperature by the relation kd = /x" 1 . To see this we
can consider the solid immersed in a simple gas ; in the
normal state they will have the same distribution-constant
for the simple reason pointed out earlier in the treatment of
mixtures of gases and of internal degrees of freedom in gas
molecules. As energy is freely interchangeable between all
molecules, gaseous or solid, there is only one variational
equation for the energy, and thus only one multiplier, ^,, is
involved for this equation in the solution by the method of
indeterminate multipliers. By means of the pressure
equation, we, as usual, identify the temperature of the gas
as (i/x)*" 1 , and, of course, in equilibrium, the solid has the
same temperature.
A conclusion of great importance follows, as in Chapter V.,
viz., the equipartition on the average of the energy between
the various components of displacement and of momenta,
^ kO for each component, this being due to the absence of
all but squared terms in the expression for the total energy.
Thus it follows directly that the total energy is 6n times this
elementary amount, or 3nk9. Hence the thermal capacity
of the solid should be 3nk. For a gram-molecule, this is
3JI where R is the gram-molecular gas-constant. This
works out about 5*95 calories per degree. This well-known
law, first pointed out by Dulong and Petit, is actually a
good approximation to the truth for many monatomic
solids, provided the temperature is sufficiently high, but the
inference that the specific heat is independent of the tem-
perature, is violently at variance with the facts. Just as in
the case of diatomic gases there arises a serious discrepancy
which classical statistical-mechanical theory has never been
able to remove. The reader may think that our simple
hypotheses are too restrictive ; but although with wider
THE SOLID STATE 125
conceptions as to the dependence of potential energy on
displacement, we would obtain a different partition of the
energy, the constancy of the specific heat as regards change
of temperature would still emerge from the treatment, and
this is an untenable conclusion, an asympotic fall of thermal
capacity to zero as the temperature approaches absolute
zero being one of most striking experimental facts dis-
covered within the last twenty years.
The removal of this discrepancy and the discovery of a
satisfactory formula for the specific heat of a monatomic
solid has, as stated, been effected by means of the quantum
hypothesis. It is time to turn our attention to this way of
escape from the various difficulties which have met us at
several points on our way hither.
CHAPTER XIV
THE QUANTUM HYPOTHESIS
14.1 The Three Stages in the History of the Quantum
Theory. The first suggestion of the momentous change
which has taken place in Theoretical Physics during the
present century, was made in 1900, when Planck, in order
to clear away a discrepancy between the experimental facts
of black body radiation and the conclusions deduced from
the current dynamical and electrodynamical theory, intro-
duced the idea that the mechanism within an atom respon-
sible for the emission and absorption of radiant energy, did
not carry out this process in the continuous manner con-
sistent with the laws of dynamics and of Maxwell's electro-
magnetic theory, but in a discontinuous and " catastrophic "
manner. It must be admitted that the reception of this
notion was rather chilly ; in the mental atmosphere of that
time anything which savoured of the " revolutionary " was
frowned on ; Einstein had not as yet arrived. But he soon
did ; in 1905, the very year which saw his first paper on the
Relativity theory, he gave decided evidence of the fact that
he always has been interested in other things in Physics
besides the theory of space and time a fact not too widely
known to the " popular " scientific public. He carried
Planck's suggestion a step further and a very " shocking "
step it was, even to those who were by that time prepared to
listen to Planck. It introduced the idea of " atomicity "
into radiation not merely in its moments of absorption and
emission by matter, b.ut also in its propagation through
space. Despite the fact that it ]gd to an immediate advance
in the study of phenomena, such as fluorescence and the
photoelectric effect, it was too much for the general scientific
world, and it is only now just as the foundations of a real
consistent Quantum theory have been laid down, that we
126
THE QUANTUM HYPOTHESIS 127
can appreciate that Einstein's " light-quantum " idea was
one of those flashes of insight vouchsafed now and then to
the man of genius. Even so, Einstein did show two years
later that Planck's suggestion in its original and less up-
setting form could be applied to the elucidation of the
theoretical difficulties which statistical mechanics encounters
in dealing with specific heats. Planck, in addition, demon-
strated a rather unexpected link between his hypothesis and
the heat theorem of Nernst (the so-called third law of Thermo-
dynamics), which was playing a great part in Physical
Chemistry at this period. Planck, however, was no " revo-
lutionary," and was also busy in recasting his original
presentation of the quantum idea, so as to soften as far as
possible the break with traditional conceptions. In 1912
Debye and Born had subjected the whole problem of the
specific heats of monatomic solids to a most searching
mathematical analysis in the manner suggested by Einstein
in 1907, and the result was a triumphant vindication of the
power of this new weapon. The trouble concerning the
specific heats of diatomic gases was also showing clear signs
of yielding to the " new treatment." The leading physicists
of the world were at last in their congresses and contributions
to journals displaying the keenest interest in the " mystery
of quantum."
The second period in this eventful history was ushered in
by three papers contributed by Bohr to the Philosophical
Magazine in 1913. In them was first propounded the theory
of the " Stationary States of an Atom," a theory which was
the. clear descendant of Planck's first form of the, Quantum
hypothesis and not the second. This period has been marked
by the development of an uneasy partnership between the
classical laws of dynamics and electrodynamics and two
postulates of Bohr's, one a flat denial of a particular result
in classical electron theory, the other, an ingenious modi-
fication of Planck's law of emission for his " oscillator."
The hostile partners have been driven in harness together,
and " made to behave " by means of another ingenious
notion of Bohr's, the " Principle of Correspondence." This
patchwork affair has, despite the incongruity of the situation,
128 STATISTICAL MECHANICS FOR STUDENTS
been at once incentive and guide during a dozen years
amazingly fertile in theoretical investigation and experi-
mental research. The debt which experimental physics owes
to this " Classical-Quantum theory" is large beyond question;
yet equally certain is it that this was not a " theory " in the
accepted sense of a perfectly consistent body of principles
from which all the essential experimental facts could be
deduced. This period closed in the autumn of 1925, when
Heisenberg, in a paper which will probably rank with
Einstein's first relativity paper of 1905, as an epoch-making
communication, pointed clearly the direction in which, we
had to go for a genuine escape from all our theoretical doubts
and misgivings. Then began the third period, a period of
the construction of a genuine Quantum theory of atomic
phenomena. Already in four years physicists can feel that
their science is based once more on a foundation which,
although not as yet complete, is as far as it goes firm and
self -consistent. To the philosophical mind it is of interest to
observe one common feature of Heisenberg's contribution
to the theory of the microcosm which we call the atom and
Einstein's contribution to the theory of the large-scale pheno-
mena of the " world." Einstein pointed out that we were
unduly hampering our ideas by trying to reconcile the
independence of the velocity of light with respect to the
frame of observation and the assumption of an absolute
space ; as the former is a physical fact, he suggested that
the latter should be abandoned and the mathematical
treatment suitably modified, and proceeded to show how
it could be done. Heisenberg also indicated the restrictive
effect on the development of atomic theory of the endeavour
to run together the idea of electron motion within an atom,
subject to the usual kinematic ideas and regarded as a
resultant of a simultaneously existing group of harmonically
related components, with the physical fact of spectroscopic
lines subject to a law of frequency so different to the law
of a series of harmonic terms ; so he surmised it would
be wiser to cease to trouble ourselves about electron move-
ments which never come into the actual field of obser-
vation. No doubt this would require a recasting of the
THE QUANTUM HYPOTHESIS 129
mathematical treatment and Heisenberg gave an indica-
tion of the way to proceed. Another feature of interest
to the scientific historian is the fact that in both instances
the pure mathematicians of the nineteenth century had
actually, without any prevision of the use the physicists
were destined to make of them, invented the suitable mathe-
matical conceptions and developed and perfected the
necessary technique ; the calculus of tensors and the calculus
of matrices were both to hand when the right moment for
their physical applications arrived, although to be strictly
accurate in our statements, both Einstein and Heisenberg had
to be informed of their good fortune by the mathematicians.
Naturally our subject, Statistical Mechanics, is being
brought into line with the Quantum Mechanics of this new
period. Just as naturally that is a matter outside the pro-
vince of a book of this nature.* The author, however, begs his
youthful readers not to be too downcast on that account.
The kind of feeling that " classical and classical-quantum
stuff " is dead and done with, and that it is just so much
waste time to bother about it is very unjustified. We travel
quickly in these days no doubt, but I doubt if any serious
teacher of physical science can see how it would be possible
to introduce the immature and youthful mind to the newer
knowledge without an adequate training in the traditional
conceptions, the manner in which they synthesised the older
knowledge, and the manner in which they were modified and
replaced by broader ideas. After all it was to deal with
difficulties in statistical mechanics that " Quantum " was
first invented, and nearly all the work of the first period was
concerned with this purpose ; and as regards the second
period, the conception of " stationary states " is still required
in order to follow the generalisation to the new formula-
tion of Quantum theory. As far as the matter of " seeing
results as quickly as possible " is concerned (which we at the
outset assumed to be the desire of the majority of readers),
there is no need to be alarmed ; the first and second periods
have provided them in abundance.
* Nevertheless, an Appendix at the end of the book will give the reader
some idea of what has happened very recently.
130 STATISTICAL MECHANICS FOR STUDENTS
14 . 2 Planck's Constant. At the time of Planck's first
suggestion with its flavour of " heterodoxy," one of the
problems agitating the minds of the physicists was con-
nected with the discrepancy between the facts of black body
radiation as discovered by an improved technique in radio-
metric measurements and the theoretical laws as deduced
from dynamical principles and the equations of the electro-
magnetic field. Thermodynamical reasoning carried the
work far enough to recognise that the formula for the
density of radiation in a uniform temperature enclosure has
a certain general character ; * in it, however, there occurs
an unknown functional form, unknown, i.e., in the sense that
thermodynamics alone can give us no information about it.
For progress towards its discovery an appeal to electro-
magnetic and dynamical theory had to be made. Un-
fortunately, if that appeal was made in a strictly " lawful "
way, the result was seriously at variance with the facts.
Planck working hard at the problem on its theoretical side,
gradually narrowed down the region in which the fallacious
step was to be found. At last he put his finger on it ; it
turned out to be the assumption of equipartition of energy
on the average between co-ordinates and momenta in any
molecular system which contribute squared terms to the expres-
sion for the energy. The italicised words are important. No
doubt there is no equipartition if that condition is not satis-
fied, but Planck was using as a model radiating and absorbing
mechanism the harmonic oscillator (the faithful ally of the
mathematical physicist, which had never yet failed him),
and its energy was a sum of squared terms. It availed nothing
to point out that after all this was a very crude model of the
radiating processes in an atom. The nature of the reasoning
was such that any conceivable mechanism following dynamical
laws should yield the proper result. But the law of equi-
partition of energy is derived from statistical-mechanical
reasoning, and that is how our subject became mixed up
with all the trials and tribulations of that period.
To appreciate Planck's hint as to how to escape from the
* Wien's Displacement Law.
THE QUANTUM HYPOTHESIS 131
dilemma, we will introduce the reader to his way of deriving
the law of equipartition. His statistical reasoning in his
first papers did not follow quite the same course as that
employed in our earlier chapters, but as a matter of fact, his
particular way of choosing complexions and counting them,
has reappeared quite lately in the literature of the subject.
Added to that, it has an interest of its own, and is, of course,
quite sound ; so it should be known to any one interested
in the elements of the subject.
Planck conceived a given number of oscillators vibrating
about fixed mean positions in an all-pervading reservoir of
energy, viz., the field of full radiation. Between the field
and the oscillators there was flux and reflux of energy. Just
as in the beginning of our statistical reasoning we had to
postulate finite phase-cells, which we afterwards reduced to
mathematical infinitesimals, so he postulated finite elements
of energy, each passing as it were, entire, and not con-
tinuously in small infinitesimal elements, at exchange
between oscillator and field. Call each element 77. Suppose
that at any moment the n oscillators have between them c of
these elements, so that their total energy is cr). How many
ways can this be done ? All the c elements might be in the
first oscillator ; represent this symbolically by af, or all in
the second represented by a/, and so on ; or c 1 of them
might be in the r th and one in the s th represented by a r c ~ l a 8)
and so on. As all the elements of energy are supposed to be
indistinguishable the number of ways of partitioning the
energy E among the n oscillators is the number of terms in
the expansion * ^
and this is known to be
w -1 C - I
(*+*-!)!
*!(*-!)! ' ' ' ' (14 ' 2 ' 1 >
>o<Y- 0! ;
Now when in equilibrium with the radiation, the oscil-
lators will have the temperature 6 of the radiation, and
possess a definite energy E. The whole system radiation
and oscillators will then statistically be in its most probable
E 2
132 STATISTICAL MECHANICS FOR STUDENTS
state. If there are N elements of energy altogether present,
the total number of complexions in this state is
/(N-c) (" + -*)' . . . (14 . 2 . 2)
nl (c 1)1
where f(x) is the number of ways of distributing x elements
of energy in the radiation, and c E/T? ; for with any one
way of distributing the N c elements in the temperature-
enclosure, there can be combined one way of distributing
c elements among the oscillators to yield one way of distri-
buting all the N elements among the various parts of the
whole system. As usual, we take the entropy of the system
in this equilibrium state to be k times the logarithm of
(14 . 2 . 2), together with a constant term which for our
purpose may be ignored as it will disappear in the differentia-
tion to be carried out presently. It is plausible to regard
the two parts into which this expression falls as the entropy
of the radiation and the entropy of the oscillators respec-
tively. So if S is the entropy of the oscillators
"
== Jclog
n\(c \)\
(n + c)\
n ! c I
= k (n + c) log (n -{- c) k c log c k n log n.
If the temperature of the enclosure be altered to 8 -f 80,
the 'energy of the oscillator system will be altered to E -f 8E
and its entropy to S + SS, an d we know from thermo-
dynamical reasoning that as no external work is done
j _ !
"SE "~ J
in the limit. But SE ^8c. Hence
l^J^S
~~ 77 dc '
= - (1 + log n + c 1 log c)
V
* We are assuming that the elements of energy are much more numerous
than the oscillators so that (n + c)/c is practically unity.
THE QUANTUM HYPOTHESIS 133
k , n + c
= -log ! .
i c
Thus - = <? 1 - 1,
c
or ?! = _J? (14.2.2)
^ gj//Jt0 j v '
This reasoning may seem extremely abstract to the
reader, who may be pardoned if he feels that the radiation
is a very shadowy kind of material to partition elements of
energy among ; nothing so tangible for example, as the
"solid" particles of our earlier "games of chance." It
would not be so to any one imbued with present-day ideas
of the essential " substantiality " of the radiation, in so far
as it possesses all the so-called mechanical properties of
matter, mass and momentum, as well as energy. Be that as
it may, he may feel reassured when he learns that the result
(14 . 2 . 2) for the average energy of an oscillator is equi-
partition of energy, if we just carry the analogy with our
former procedure to its logical limit, viz., assume that the
elements r\ are mathematical infinitesimals, as we formerly
assumed the phase-cells, in terms of which we defined com-
plexions and states, to be infinitesimals in our final calcu-
lations. If we do so, the right-hand side of (14 . 2 . 2) is equal
to
which approaches the value kO as t\ approaches zero. And so
the average energy of the oscillator becomes kd, and this
agrees with our former result, | kO of kinetic, and J k0 of
potential on the average.
However, it was the really brilliant idea of Planck just to
refuse to go to the limit. With a flash of insight he saw that
in refusing to do so, and in thus denying the equipartition
law (as we shall see presently) lay salvation. Here We must
take on faith a result which concerns the electrodynamical
side of Planck's complete argument as distinct from the
purely statistical with which we have been immediately
134 STATISTICAL MECHANICS FOE STUDENTS
concerned. In that he had deduced the relation between
the energy-density of the radiation and the average energy
of an oscillator in temperature equilibrium with it. If he
took the latter to be k6, the result was at variance with
experiment ; so he took (14 . 2 . 2) as it stands with T? finite,
and considered the conclusion to be drawn from that. At
once he saw that in order to satisfy the general character of
the formula for the energy-density, derived from purely
thermodynamical reasoning and referred to on page 130,
77 would have to be proportional to the frequency of the
oscillator and the constant of proportionality would, more-
over, be a universal constant. So calling the frequency v he
wrote
7] = JlV,
and so
5= ^ . . . (14.2.3)
n exp (hv/k0) I
From this could be derived at once a formula for the energy
of full radiation as distributed among its various frequencies.
Within a short time the experimental physicists were con-
vinced that it was correct, and derived from the observations
a value for h which has since been confirmed in several
experimental researches inspired by other applications of
the quantum idea. The accepted value for h is
6-55 x 10~ 27 erg-seconds,
for, as will appear at once, its physical dimensions are the
product of energy and time, i.e., the dimensions of action.
14 . 3 The " Quantised Paths " of an Oscillator. We have
given Planck's original treatment (somewhat amplified) of
the statistical side of the problem. It will be interesting to
deal with the matter in a manner more in keeping with the
methods which we have used hitherto, now that we appre-
ciate Planck's break ,with tradition. This will have the
added advantage that it will give us also a truer idea of
where the " quantisation " is really situated. In terms of
the conceptions of section (14.2) we naturally speak of
" quanta of energy/' but if there is any idea latent in the
reader's mind that this must imply discrete " atoms " of
THE QUANTUM HYPOTHESIS 135
energy preserving an identity through all vicissitudes, as
we have been accustomed to assume for atoms of matter,
he must disabuse his mind at once of this idea. Planck
himself would have none of it in those days and when
Einstein, in his study of the photoelectric effect, propounded
quantum views savouring very much of such heretical
notions, Planck protested vigorously, and recast his whole
presentation in such a manner as to bar out this idea, which
was really at that time repugnant to the sense of con-
tinuity produced by the theory and the experimental facts
of the propagation of radiation. No doubt the oscillator
could only take in " lumps " of energy, hv, or get rid of them
if it emitted or absorbed at all. The reader will realise that
the whole of the argument would be upset, if there could be
fractions of 77 in an oscillator as well as integral multiples.
But once out, the energy merged into the continuous field.
Nevertheless, one conclusion could not be avoided ; between
emissions and absorptions the vibrations of the oscillator
could only be executed with one of a discrete series of
amplitudes ; i.e., it could only exist in one of a discrete set
of " quantum states." We are familiar in ordinary mechani-
cal reasoning with the notion that a harmonic vibrator can
be given any of the infinite number of amplitudes between
zero and some upper limit. But that notion will not serve
here. We can very readily select those " quantised " ampli-
tudes, by means of the information already obtained.
In the symbolism of section (5.1) the energy of the oscil-
lator is
.
or +
2a 2
If this has a constant value, e, the representative point of
the oscillator in a phase-diagram will be on the ellipse
2 6/6 2 e a
Its semi-axes are (2 /&)* and (2 ea)*, and its area is the
product of these by TT, i.e., 2 TT e (a/6)*.
136 STATISTICAL MECHANICS FOR STUDENTS
But (6/a)* is the pulsance, 2 n v, of the vibration, and so the
area of the ellipse is
- (14.3.1)
V
In consequence, if in this quantum state the oscillator holds
r " quanta of energy of frequency v," its elliptical phase-
path has the area
rh (14.3.2)
Thus the discrete quantum paths of the representative point
in the phase-diagram of the oscillator are a series of similar
ellipses separated from each other by elliptic annuli of area
h, the lowest quantum condition being represented by the
origin. Thus no matter what the value of v is, i.e., no matter
what is the value of the quantum of energy, the phase-
diagram is divided up into areas always of the same magni-
tude. The only quantum which hab so far any claim to
atomicity is thus the " quantum of action," for it is action
which is represented by an area in the phase-diagram. It
will be wise to remember that a quantum state is not a
static condition ; the oscillator vibrates to and fro, and the
representative point keeps rushing round the appropriate
ellipse between the ' c catastrophic ' ' emissions and absorptions .
The statistical problem is now easily dealt with in our
more customary manner. A complexion of the system of
oscillators is determined by the manner in which we assign
individual, identifiable points to the various ellipses ; there
will be some outer limit settled by the whole energy of the
system. The ellipses take the place of the phase-cells in our
previous arguments; an interchange of points between
ellipses alters the complexion, but not the statistical state ; *
a mere shifting of points along the ellipses does neither.
The number of ways of assigning n points to the origin, %
to the first ellipse, n 2 to the second, and so on, is
nl
n. n. n 9
* Do not confuse the two uses of the word " state " from this point
onward. We speak of a " quantum state " of an oscillator, and, later, of
an atom ; " Statistical state " still refers to the whole system of oscillators
or atoms.
THE QUANTUM HYPOTHESIS 137
We then proceed as before. The logarithm of this is varied
and the variation put equal to zero. The conditions for con-
stant total number and total energy introduced, and we
arrive at the usual result. In the most probable state the
number of oscillators in the r** quantum state is
C e" w
where r = rhv.
The constant //, is, as usual, identified with (k6)~ l ; for we
are really considering the oscillators now as denizens of gas
atoms in the manner of Chapter V.
To determine C, we have
C (1 + e~^ + e~^ hv + e~ 3 ^ + ...... ) = n*
or C = n (I e-"*") . . . . (14.3.3)
The whole energy in the oscillators is
C 2 T e-*"'
ro
which is equal to
C hv (e~* hv + 2 e~ 2 * hv + 3 e~* v + ...... )
= rihv (1 e-* hv ) e'"** (1+2 e~^ hv + 3 e~^ hv + ...... )
nhv
e hv I
or the average energy of an oscillator is
hv
as before.
It will be seen that in the most probable statistical state of
the system, the majority of the oscillators are in the lowest
quantum state represented by the origin, the number in the
next quantum state is obtained by multiplying the former
* The unity in the series arises from the lowest quantum state, r = o.
138 STATISTICAL MECHANICS FOR STUDENTS
by exp ( hv/kd), in the next by exp ( 2 hv/k6), and so
on. If the temperature 6 decreases the multiplier e""^",
becomes smaller and smaller approaching zero as 6 approaches
zero. Thus as the temperature falls, the oscillators tend to
crowd into the state of zero energy. If 0, on the other hand,
increases, the exponential factor increases gradually having
unity as its limit when 6 is infinite. Thus there is with
rising temperature a tendency towards more equal distri-
bution of the oscillators among the quantum states.
14.4 Planck's Alternative to the Strict Conception of
Quantum States. It is possible that the reader may not
quite realise the point of the extreme hostility to Planck's
views at the time. The assumption of quantum states seems
harmless enough in its way, and the statistical argument,
at all events in the form outlined in section (14.3), appears
as unimpeachable as it was in earlier applications. That is
true enough ; it was not the statistical part of his investiga-
tion which failed to command general assent at first. But
there was another part just as necessary to the argument,
and in that part Planck assumed that the amplitude of the
oscillator could vary in a continuous manner. The oscil-
lator is subject to the electromagnetic forces in the radiation
and those forces which have the same frequency as the
oscillator, or frequencies relatively near, impress an oscil-
lation of increasing amplitude on it an illustration of the
well-known phenomenon of resonance. The safeguard
against undue heaping up of energy in the oscillator is its
own radiation due to the accelerated motion which its
vibration implies. The balance between absorption and
radiation is required to keep it in an average condition as
regards energy, whether that condition be equipartition or
any other. The mathematical equation which expresses
that balance leads to the relation between the density of
each constituent of full radiation and the average energy of
oscillators with the corresponding frequency. Thus it will
be seen that this side of the argument was absolutely
essential to Planck's final result, and in working it out he
had to assume the continuous increase or decrease of the
energy in the oscillator. In an endeavour to minimise the
THE QUANTUM HYPOTHESIS 139
contradiction as far as possible, and as a protest against what
he considered to be an unnecessary extension of his idea, he
recast the statistical argument so as to suit at all events the
assumption of continuous absorption, although he still had
to keep discontinuous emission in it. Though it has not
much place in the general exposition of quantum physics,
as it developed later, it should be known to the student of
these matters, as it suggests one conclusion which may
possibly be justified soon as an experimental fact. To
appreciate its point of difference with the first method, let
us conceive that the phase-diagram is partitioned into phase-
cells by drawing equal energy ellipses very near to one
another. The phase-cells are the elliptic annuli, each of
area Se/v where Se is the step of energy between one ellipse
and the next. By classical methods the density of points
in an annulus is proportional to e~^ and the number in it
is D e~^ e Se/v where P is a constant given by
Dr
__ e-^de =n . . . . (14.4.1)
VJ
so that D = Ufjiv. In obtaining this, we have, as formerly,
narrowed down the cells to infinitesimal dimensions in the
last resort. In the equilibrium state as many representative
points pass outward in a given time across any given ellipse
(absorption) as pass inward across it (emission), and an
inward, or outward journey could start from any point.
But in Planck's second form of his theory, an inward journey
could not start from any point ; once a representative point
has passed out beyond one of the critical ellipses of area
A, 2 A, 3 A, etc., it cannot move inward ; not until it has
reached the next critical ellipse is that possible. In short,
emission is only possible at certain critical amplitudes. At
such critical moments of its history the oscillator may
radiate or it may pass on into the next " zone of safety "
free to gather up energy for another spell without danger of
loss. But some of the oscillators are bound to radiate when
they reach any critical ellipse ; otherwise the balance
between absorption and emission over relatively long periods
would not be maintained. When one speaks of an " outward
140 STATISTICAL MECHANICS FOR STUDENTS
journey/' one visualises the representative point on the
phase diagram executing a spiral path gradually widening
out from the origin ; such a path may terminate suddenly
at the first critical ellipse and restart at once from the
origin, or it may continue widening out uniformly in the
annulus between the first and second ellipses until it reaches
the latter, when it may still continue or suddenly stop and
make a fresh start from the origin ; and so on. Thus the
number of points in any h annulus is at any moment less than
in the one just inside it and more than in the one just out-
side ; but just how much less and how much more ? Well,
one can, as before, consider these h annuli as finite cells, and
assign n l points to the first, n 2 to the second, and so on, and
proceed to work out the most probable state ; but in writing
down the constant energy condition we realise that all the
points in one of these cells do not correspond to the same
energy ; we cannot assume that, as we are not ultimately
going to make these cells infinitesimal in size. A plausible
proposition is that the points in an h annulus are uniformly
distributed over it, and in that case when we write the
equation
e r is the mean energy of the r th annulus, i.e., it is equal to
(r ) hv. The usual result emerges
n r = C e-^ r
where
<r = (' - i) *"
As usual C is determined by
C e-^r = n,
r=l
or
C e-f (1 + e-* + e~ 2x + ...... ) = n,
where x is for the moment written instead of phv.
Thus
C = n e 2 (1 O-
THE QUANTUM HYPOTHESIS 141
The whole energy E is given by
E = C 2 r e-"*'
3 e~
= ~
+ 2 (e~ x + 2e- 2 * + 3 e~ 3x
nhv . , ., ~ x
= - + nhv (1
2 ' v ' (1 -
_
___ j
2 e* - I
Thus the average energy of an oscillator is now
hv + hv n4 4 2)
exp (hv/k0) 12'''*
There is an interesting difference between this result and
(14.2.3). If we make 6 gradually approach zero, the expres-
sion (14 . 4 . 2) does not approach zero but hv/2 as its limits ;
thus at absolute zero the system of oscillators would have an
energy of amount nhv/2. Planck could still make this new
expression yield the satisfactory expression for black body
radiation, and he was also able to use the new model in an
explanation of the (then) puzzling features of the photo-
electric effect without recourse to Einstein's light-quanta.
This being so, he felt justified in asserting that here there
was theoretical evidence of the possibility that at absolute
zero of temperature matter is not quite devoid of energy.
The existence of " nul-point energy " is suggested.
The number of oscillators in one of the finite energy ranges
defined by the critical states of emission bears to the number
in the range just below the ratio eT x to unity. Thus the
chance that an oscillator will radiate when it reaches a
critical state is 1 e~ x , provided it radiates the whole
amount which it contains and restarts a fresh accumulation
of energy from that condition. In his statistical-electro-
dynamic argument, Planck assumes that emission has this
142 STATISTICAL MECHANICS FOR STUDENTS
character, and introduces a definite postulate about the
chance of emission in a critical state ; these hypotheses
correspond to the assumption made above in this simpli-
fication of his argument, that the points in an h annulus are
equally distributed. The reader will observe that in his
first exposition emission had not of necessity this character ;
the representative point could jump about from ellipse to
ellipse, in its changes of state, inward or outward ; if inward,
the leap was not necessarily right to the origin.
CHAPTER XV
THE THEORY OF THE STATIONARY STATES OF AN ATOM
15.1 The " Action-Integrals " of a System. Notwith-
standing Planck's conservative tendencies the movement
which he initiated took his first idea as its guide for further
development. To see how this came to pass, we must deal
with some general dynamical features of more complex
systems than a simple harmonic oscillator ; it is too crude
a model for further exposition. For one thing the assumption
that even for an oscillator with but one degree of freedom
the period is independent of the amplitude, is not really
true of the many physical systems which the oscillator is
taken to represent in elementary statements the pendulum
for instance, or the body bobbing up and down at the end of
a spring. But if this is so, the apparent simplicity of
quantising by means of the energy content disappears, as
that clearly depends on a constant frequency. But we saw
that another way of declaring how to select the quantum
states of the harmonic oscillator is to say that the choice
will fall upon those amplitudes for which the accumulated
action in one period is once, twice, thrice, etc. the value h.
This choice is still open to us, and during the second period,
referred to in section 14 . 1, it was the keynote to the whole
situation.
The restoring force which controls the oscillator not being
proportional to displacement beyond minute values of the
latter, it turns out, as stated, that the frequency of the
oscillation depends on the amplitude. The oscillator will,
however, in any actual vibration, have a definite momentum,
p, for a given value of displacement, q. (It must be clearly
understood that " damping " is excluded from consideration.)
The integral of pdq can be calculated from the knowledge of
the particular law of force, and if this integral is calculated
143
144 STATISTICAL MECHANICS FOR STUDENTS
for the complete period which exists for this given vibration,
we shall denote its value by J. This accumulated action in
one period is usually termed the " action-integral " of the
system. It is like an integration constant occurring in the
statement of q as a function of the time ; it is a " constant "
during any movement involving a given amplitude, but
varies in value if the swings are altered so as to involve a
wider or narrower range. Amplitude, frequency and energy
can all be expressed as functions of J. It will be recalled that
in the case of a harmonic oscillator, if the energy of a par-
ticular vibration is e, then the accumulated action in one
period is e/v, so that the quotient of the former by the latter
is the frequency, or to put it another way, any finite change
in the energy if divided by the change in the action-integral
accompanying it, gives the frequency of oscillation. The
result for an " anharmonic " oscillator is a generalised form
of this. Let E(J) be the function of J, which is equal to the
energy ; E(J + SJ) is equal to the energy for a slightly
different amplitude with an action variable J + 8 J. It can
be proved that on dividing E(J + SJ) E(J) by SJ and
going to the limit, we obtain the frequency of the oscillation
with the action-integral J. In short, the frequency of
oscillation is d^(J)/dJ. In addition to this modification of
the results it must be borne in mind as well that we cannot
write
q = A cos (cot e)
where
*E(J)
W - 2 "-d3-
The oscillation is not a simple harmonic one. Instead it is
known that the correct result is
q = Aj, cos (cot <j) -f A 2 cos (2 cot < 3 ) + A 3 cos
(3 oot - < 3 ) + ......
where ^ 1? < 2 , </>& the epoch-angles, and A lf A 2 , A 3 ,
are known functions of a second integration constant
and J.* The selection of the quantised oscillations for this
* The equation of motion is of the second degree, and in its complete
solution two integration constants must make their appearance.
STATIONARY STATES OF AN ATOM 145
oscillator is, if we adopt the method used for the model
atoms of Bohr's theory, made by choosing those whose
action-integrals are whole-number multiples of h. To be
strictly accurate it was discovered as time went on that
some experimental results were more easily brought within
the bounds of the theory if this selection were interpreted a
little more loosely and " half -quantum numbers " allowed,
so that the action-integrals might be put equal to rJ or
(r + |)J where r is a positive integer. As a matter of
interest this extension of the quantising principle does for
certain systems emerge quite naturally from the most recent
formulation of Quantum mechanics. It should, however, be
noted carefully that this extension does not bring in its wake
any necessity for " half -quanta of energy." The reason why
will appear presently.
To go on to a system with two degrees of freedom, such as
the ordinary pendulum might be considered to be, an
interesting feature of the vibrations of such a system can
be observed very readily by giving a pendulum a slightly
elliptical swing. It soon appears that the orbit of the bob
is not an ellipse, nor indeed any closed oval curve. Roughly
we say that the plane of vibration of the string is rotating
round. Strictly, of course, there is no such plane. Another
mode of expression is to say that the bob is going round an
elongated ellipse, whose long axis is slowly " precessing " in
a horizontal plane in the same sense as the motion of the
bob round the orbit. What is really happening is that the
bob does not return to the same place at its places of greatest
and least distance from the centre. This arises from the
fact that it has really two distinct periods of vibration, which
may be nearly equal but are not exactly so. The amplitude
one way is quite large ; at right angles to that way quite
small. It will happen, of course, that if the two periods have
values which are commensurable then ultimately the curve
after many convolutions will " re-enter " into " itself," and
the body proceed to execute the same path once more. But
if the periods are incommensurable, this is not so. The
simplest way of bringing out the two periods is to realise
that there is a certain interval during which the radius
146 STATISTICAL MECHANICS FOR STUDENTS
vector from the centre changes from its greatest value to its
least, then to its greatest, once more to its least, and finally
to its greatest. This is one period, the period of " libration "
of its radius vector. (We are really at the moment thinking of
a vibration truly in a horizontal plane; actually the bob of a
pendulum travels on a spherical surface.) On the other hand,
the angle which this radius vector sweeps out in the horizontal
plane, increases by 2 77 in a different time ; in this time cos
and sin 0, where is this " azimuthal " angle, librate in value
between the maximum and minimum values, + 1 an< l 1?
and back again. In this system the accumulation of the
action goes on in two ways. The vibrating body has a
certain angular momentum round the origin, mr*8, where r
is the radius-vector to its position on the oval path ; if we
integrate mr 2 9d0 between and 2??, we obtain one action-
integral J 1( The body also has a linear momentum m'r to
and from the origin ; if we integrate m'rdr throughout the
period of radial libration, we obtain a second action-integral
J 2 . The energy in any prescribed orbit can be found from
a knowledge of its two action-integrals ; if E(J 15 J 2 ) is the
function which is equal to the energy, then the two fre-
quencies azimuthal and radial are
1? J 2 ) , SEfa, J 8 )
,
d
X 9J 2
respectively. That is known from dynamical theory. The
expressions for r and in terms of t take the form
oc oo
r == 2 H A r8 cos (r^ + sa> 2 ) t + <j> rs
:== > 2j B cos ] (I*CL)I -f- 5cL)o) t -f- <p o - .
r.-r= oo = oo I '
Here the A r5 , B r ^ and ^ are functions of J 1} J 2 and two
other integration constants, and as stated, aj l = 2-rr SE/SJj,
C0 2 = 277 3E/9J 2 .
We have here an example of a " conditionally -periodic "
system. The system may not apparently be periodic ; for
we have seen that if o^ and co 2 are incommensurable, the
STATIONARY STATES OF AN ATOM 147
vibrating body never really returns to any former position,
but the solution shows how the periodicity is latent in
artificially separated 'parts of the motion.
The suggestions for quantising in this case follow the same
lines as before ; the quantised paths are chosen so that J l =
r-Ji and J 2 ~ rji where r l and r 2 are positive integers (or
numbers such as r + ^ may be involved).
15 . 2 Bohr's Postulates for the Atom. Possibly the
reader may now begin to have some idea of the quantisation
of orbits, and how action and Planck's h constant are in-
volved in it. Taking the hydrogen atom with its single
electron, it has to be observed that the strictly elliptic orbit
of the electron round the nucleus as a focus, deduced from
the simple inverse square law of attraction, is too ideal.
Owing to a variety of causes, the forces caused by neighbour-
ing atoms, or external fields imposed by an experimenter in
the study of the Zeeman and Stark effects, or the relativity
change of mass in the electron due to changing speed in the
orbit, the orbit is not really re-entrant. Just as we have
pointed out in the case of the two-dimensional anharmonic
oscillator, there are two periods, one involved in the libration
of the radius vector between its extreme values, one in the
increase of the azimuthal angle by 2??, and a repetition in the
values of its circular functions. Two action-integrals are
involved just as before, and one of Bohr's fundamental postu-
lates consisted in assuming that, despite the deductions of
classical dynamics, the orbits whose action-integrals are equal
to integral multiples of h have an inherent stability, non-
dynamical to be sure, but physical in the sense that the
assumption could be used with great effect to unravel for
the first time some of the intricacies of spectroscopic obser-
vations which had hitherto baffled physicists. The remaining
non-quantised paths, just as dynamical and from the point
of view of Dynamics just as "unstable" as the quantised,
are irrelevant to the explanation of the physical facts.
Conceivably, if undisturbed, the electron could remain in a
quantised orbit for ever, but if disturbed, it would have to
find a new semi-permanent home in another quantised orbit,
not in any of the mechanically possible, but " quantically "
L 2
148 STATISTICAL MECHANICS FOR STUDENTS
impossible orbits. The atom has a discrete number of
"stationary states/** It should be mentioned that
Sommerfield and Schwarzschild gave considerable assistance
at the outset on the mathematical side in applying the
quantising rules to particular problems.
The second postulate of Bohr concerns the " quantum-
jumps " during which the atom leaves one stationary state
and enters another. In Planck's early oscillator theory the
quantum of energy is determined by the frequency of the
oscillator vibration ; h if multiplied by the latter gives the
former. In view of the multiplicity of kinematic frequencies
now involved, no success in that direction seems possible.
Bohr's ingenious modification lay in dividing the change of
energy between two states by h to obtain the frequency of
the emitted or absorbed radiation. Optical frequencies
ceased to be identified with kinematic. Terra firma began
to appear in the region of spectroscopy where, in the word
of the late Lord Rayleigh, there had been formerly a " bog."
We have assumed that the orbit of the electron is in one
plane ; virtually that limits the degrees of freedom to two ;
two co-ordinates are sufficient to determine the position of
the electron ; there are two distinct action-integrals for any
orbit, classical or quantum, and in consequence two funda-
mental frequencies involved in each orbit. But the degrees
of freedom are really three ; our limitation of the orbit to
one plane has concealed that fact. But if we apply some
external force to the electron, such as a magnetic field, the
normal to the plane of the orbit precesses round a line
through the nucleus parallel to the field, just as the axis of
a spinning top precesses round the vertical. To define the
electron completely now we must have a co-ordinate
defining the plane of the orbit or the normal to it, in addition
to the radius vector and azimuth of the electron in the orbit.
This can be chosen to be the angle made by a plane contain-
ing the normal to the orbital plane and the field line through
the nucleus with any reference plane containing the latter
line. This precession involves additional energy ; and also
* Note that *' stationary " does not mean " static." There is plenty of
movement in a stationary state.
STATIONARY STATES OF AN ATOM 149
a new angular momentum round the field line. Prom the
latter, using dynamical laws, can be calculated a third
action-integral equal to the amount of action accumulated
in this fashion in the period of precession, the third period
of the system. Classically this may have any value ; the
requirements of the Quantum hypothesis limit it again to
integral multiples of h. This amounts to stating that the
orbit can only have a set of discrete orientations with respect
to the external influence. Hence arises " space-quantisa-
tion," and a third quantum number enters in the selection
of stationary states.
Nor is this all. In our solar system the planets spin on
their axes as well as rotating round the sun in their orbits.
So in recent years the " spinning electron " has come along
to give us a hand in this entertaining puzzle of defining
stationary states so as to conform to the spectroscopic, the
magneto-optic, the electro-optic and the thermal evidence.
This involves another angular momentum and period ;
another action-integral and a fourth quantum number.
All this has been written in connection with the single-
electron atom. As a matter of fact, in view of a special
feature of Bohr's theory of optical spectra, it holds good in
an approximate way for much more complicated atoms.
Further we can hardly go without transcending the limits
of space and possibly the reader's ability to follow the
necessary analytical statements. In the last resort, if there
are I electrons, we can (regarding the nucleus as sufficiently
massive) think of the system as having 31 degrees of freedom.
The radial vectors of the electrons will librate between
maximum and minimum values, the angles necessary for
co-ordination will also librate or their circular functions will.
These are necessary conditions of periodicity and stability,
even if they are not sufficient. The whole theory can be
worked out under certain definite mathematical assump-
tions. There are 31 action-integrals involved, and 3Z fre-
quencies given by the same rule as before
150 STATISTICAL MECHANICS FOR STUDENTS
Quantisation chooses those states of orbital motion in which
the J r are whole-number multiples of k, and so each station-
ary state corresponds to a definite set of 31 integers. In fact,
in recent theory less and less interest is being shown in the
attempt to make pictures of the orbits or give analytical
expressions for them. The state is determined by its quan-
tum numbers. If we know the energy as a function of the
quantum numbers, i.e., of the action-integrals, the change
of energy in a jump from one state to another can be cal-
culated, and the frequency of the emitted radiation obtained
by Bohr's second postulate. The results must, of course,
agree with spectroscopic evidence ; that acts as a check on
any hypotheses we introduce for formulating the energy in
terms of the quantum numbers. For that purpose the model
atoms of the older period with their electron orbits have
still their uses, but the clear cut planetary picture seems at
the moment to have the same doom confronting it as the
ether not so much actual denial as mere apathy. Useful,
perhaps, as mental helps for those without the necessary
mathematics, but beyond that of little use.
CHAPTER XVI
DISTBIBUTION OF A SYSTEM IN ENERGY
16.1 Quantisation and a priori Probability. When dealing
with complexions and statistical states of a molecular system,
we introduced as a fundamental hypothesis the statement
that all complexions have the same a priori probability which
involves in its turn the assumption that the representative
point of a given molecule is as likely to be in one phase -cell
as another. That of necessity implies that the phase-cells
have all the same magnitude. Similarly in dealing with
Planck's oscillator and deriving his expression for the
average energy in section (14 . 3), we obviously assumed that
an oscillator is as likely to be on one quantum path as
another ; these paths take the place of the equal-sized phase-
cells in the classical treatment of non-quantised systems,
and again all complexions have the same a priori probability.
So long as a system does not involve both quantised and non-
quantised elements, no necessity to fit the two aspects
together arises ; nor is it even troublesome to deal with a
system of complex molecules whose positions and trans -
latory movements follow classical laws and whose internal
oscillations follow quantum laws.
But suppose we have to deal with the ejection of electrons
from atoms in the photo-electric effect or ionisation caused
by collisions or X-rays ; or dissociation where atoms are at
times bound as parts of larger particles and at times are free
and independent particles themselves ; or sublimation of
molecules from a solid state. In these examples we see the
possibility of a system being composed of particles any one
of which may be in a quantised path sometimes and in a
non-quantised path at other times. In the latter condition,
division of a phase-diagram into phase-cells is the suitable
machinery for counting complexions ; in the former not so ;
151
152 STATISTICAL MECHANICS FOR STUDENTS
quantum paths must be used. How are we to combine the
two ideas ? Even if we regard all quantum paths as equally
likely between themselves, and all phase-cells as equally
likely among themselves (i.e., that a particle is as likely to
be in one phase-cell as in another if it is behaving in a non-
quantum way, and on one quantum path as another if
behaving in a quantum way), what is the chance that a
particle will be in a given phase-cell as against the chance
that it will be a given quantum path ? Clearly the answer
will depend on the size of the cell ; the smaller it is the less
the relative chance. The following procedure has been
adopted on the grounds of its plausibility, and no facts are
known with which it is at variance. Recalling as a simple
illustration the case of Planck's oscillator, we see that a
given quantum path concentrates on itself as it were, all the
representative points which in classical conditions would be
dotted about in the elliptical annulus between it and the
next path or rather in the adjacent halves of the two neigh-
bouring annuli which it separates. This area, whose points
are thus swept into one linear channel, is h. Now if s is the
area of a phase -cell, and S the area of the whole phase -
diagram determined by the volume and total energy of the
whole system of oscillators, then the a priori probability
that a representative point, if classical motion were involved,
would be in a given phase-cell, would be s/S. So it seems
plausible to assume that the a priori probability that a
representative point shall be on a given quantum path is
h/S. Thus the relative probabilities for non-quantum and
quantum possibilities is s/h, or, if we wish to avoid bringing
in A priori probabilities explicitly, and content ourselves as
hitherto, by counting complexions and making the number
stand for the relative probability of a statistical state, we
must assume that s is equal to h, i.e., choose the phase-cells
suitable for complexion-counting in non-quantised motions
to have the size h.
This is for a system whose particles have one degree of
freedom and whose phase-diagram is partitioned into areas
which represent action. For a system whose particles have
the usual three degrees of motion, it is a natural generalisa-
DISTRIBUTION OF A SYSTEM IN ENERGY 153
tion to take as the size of a phase-cell for non-quantised
motions A 3 , and then to regard the representative point of a
given particle to have the same A priori probability of being
in non-quantised motion in a given phase-cell as of being in
quantised motion on a given quantised path. Alternatively,
if p is the a priori probability of the point being in six-
dimensional phase-cell of size s, the a priori probability that
it is on any quantised path is p h*/s. Will the reader please
bear in mind that in this we have not been considering the
particles as having internal degrees of freedom themselves ?
They have been regarded as simple structureless particles
capable of flying about in certain parts of the enclosure,
just like the constituents of a gas, or, on the other hand,
being " bound " to some centres of attraction, in other
parts, and so probably subject to quantum conditions. The
question of complex molecules does not involve so seriously
the doubt we have referred to. The phase-diagram will then
have a higher dimensionality than six ; those extra dimen-
sions which are required to deal with internal motions will,
as far as we can judge, involve quantum methods through-
out. There is a difference between the simultaneous exis-
tence of quantum conditions and non-quantum conditions
for different degrees of freedom, and the alternation of the
same degrees of freedom between periods of quantum
motion and periods of non-quantum motion.
16 . 2 Energy- Hypersurf aces and Energy-Shells in the Phase
Diagram. In dealing with the structureless particles of the
previous section, nothing has been said as regards the
*" shape " of the phase-cells. In earlier chapters it has been
understood that the cell is analogous to a rectangle in a two-
dimensional phase-diagram ; i.e., it is looked upon as an
extension-in-phase, such that the co-ordinates and momenta
corresponding to any phase in it are respectively greater than
some set of values x, y, z, , 7?, , and respectively less than
* + Sx, , + 8, where Sx, S are small
increments. But there is no compulsion to adopt this view.
The fact that in the most probable state the numbers in
each cell involve e" 1 " as a factor suggests an alternative
method of delimiting the cells which will prove convenient
154 STATISTICAL MECHANICS FOR STUDENTS
at times. The energy of a particle is given by the function
( 2 + f] z + 2 )/2w, at all events in the absence of an external
field (and ignoring also for the moment any internal energy
or energy of rotation). All phases whose energies are
individually less than are bounded by a region in the phase-
diagram, such that
fa + ,a + a = 2m . . . . (16.2.1)
Its six-dimensional magnitude is determined by inte-
grating dx dy dz d drj d throughout the volume of the
system of particles as regards x, y, z, and throughout a
sphere determined by (16 . 2 . 1) as regards , 17, . The result
is
t 7 (2m )*v . . . . (16.2.2)
*$
where v is the volume of the system. The phases which
satisfy (16.2.1) exactly, have a certain extension-in-phase,
but it should be realised that it has not the full dimen-
sionality (six) of the phase-diagram. Just as in ordinary
geometry, the points whose co-ordinates satisfy an equation
such as
f(x, y, z) = c,
have an extension which is only superficial, and do not
occupy a three-dimensional volume, so the extension of the
phases satisfying ( 1 6 . 2 . 1 ) is only five-dimensional. We call
(16 . 2 . 1) a " hypersurface." * Apart from names, however,
we realise that on strictly classical lines the chance that the
phase of any particle in the system satisfies (16.2. 1) is zero.
But if we consider another hypersurface
2 +i? 2 + a = 2m( + 8c) . . (16.2.3)
the phases which satisfy any equation
P + i* + ? = c,
where c has any value between 2 m and 2 m (e + Se)
occupy a six-dimensional region in the phase-diagram, and
* Of course (16 . 2 . 1) would represent an ordinary surface in a simple
three-dimensional momentum-diagram. In the phase -diagram, however,
six dimensions are involved. The absence of x, y, z in ( 16 , 2 . 1 ) shows that
the hypersurface has " cylindrical " properties.
DISTRIBUTION OF A SYSTEM. IN ENERGY 155
the chance of a particle in the system having one of these
values is not zero ; it is proportional to the size of this
"energy-shell lying between" the hypersurfaces (16.2. 1)
and (16 . 2 . 3). The picturesque phrase " lying between "
has a convenient brevity ; its meaning is quite definite, and
can be expressed analytically as above. The size of this
shell can be obtained as the differential of (16.2.2); it is
27r(2m)He i S(: . . . . (16.2.4)
Shells such as these lying between adjacent members of
the family of hypersurfaces (16.2. 1) for varying values of e
can be selected as phase-cells. In the most probable state
of the system of structureless molecules the number of
particles whose phases lie in such an energy-shell is
Cve-KJS* . . . . (16.2.5)
where C is determined by
r
Jo
Cv e* e~*" dt = n.
Jo
When we pass to the question of internal degrees of free-
dom, the matter can be treated in an analogous way.
Leaving aside in its turn and for the moment the question
of the general position and motion of a molecule, suppose
the internal structure of the molecule to be defined by
values of certain co-ordinates q^q^ , ?/ and momenta
p l9 p 2 , pj. These can at the moment, pending a
fuller account of the dynamics of such systems, be thought
of as a suitable number of Cartesian co-ordinates or polar
co-ordinates of sub-particles with reference to origin and
axes in the molecule, and the linear momenta or linear and
angular momenta accompanying them. The salient point
to bear in mind at present is that the product of any co-
ordinate and its corresponding momentum has the physical
dimensions of action (energy X time). The internal energy
of a particle will be given by some function of the q r and p r ,
<(?i> ?/, Pi, , p f ), or briefly <(#, p). In the
2 /-dimensional phase-diagram, the phases satisfying
>) = c (16.2.6)
156 STATISTICAL MECHANICS FOR STUDENTS
occupy a 2/ 1 dimensional extension. The 2 /-dimen-
sional region which " lies between " (16 . 2 . 6) and
*(<!, #) = * + . (16.2.7)
contains those phases which satisfy any equation such as
4(9, P) = c
where c has a value between e and + Se. It will have a
magnitude
X() ^
where x(e) is & function of e, involving of course such con-
stants as are present in the functional form of (f>(q, p). In
the most probable state of the system of molecules the
number whose internal co-ordinates and momenta will at
any moment lie in the energy-shell between (16 . 2 . 6)
and (16 . 2 . 7) are given by an expression involving
e~>"x(e)Se.
16.3 Energy-Hypersurfaees and Paths. Following the
representative point of any particle, this will travel about
the phase-diagram in a fortuitous sort of way. For a time it
may remain on an energy-hypersurface ; it is experiencing
no influence from other particles or radiation. But at times
its path will be stretched from some point on one hyper-
surface to a point on another ; it is then under such influences.
Let us consider a little more closely the paths when the
particle is undisturbed and still, for convenience, confine
ourselves to the internal motion. The co-ordinates q and
momenta p can theoretically be determined from the
equations of motion. They are then expressed, each as a
function of the time t and certain constants.* Of these
constants a certain number will depend on the internal
structure of the molecule ; we shall denote them by
a l9 a 2 , Others, 2/, in number, are introduced in the
integration of the equations of motion. These integration
constants we shall denote by b l9 6 2 > &2/- We then
have
* Chapter XXIV. will give a fuller explanation of these statements which
may be accepted now without question.
DISTRIBUTION OF A SYSTEM IN ENERGY 157
q l = B l (t, a, b)
(16.3.1)
q f = f (/, a, b)
Pi = 0i (^ , 6)
jp, = 0, (J, a, 6)
where flj ( ), fa ( ) are 2/ functional forms. If we
give a definite set of values to the b constants, then the sets
of values acquired by q v , PJ, as we vary t from
oo to +00, constitute a path of the particle in undis-
turbed motion. The b constants vary from path to path.
But the a constants have the same values in all paths ; they
are characteristic of the molecule itself, and its particular
structure. We represented the energy by <(#, p). If in this
function we substitute for the q r and p r the functions O r ( )
and iff r ( ), the terms involving t must vanish identically,
since the energy of the particle is constant in undisturbed
motion, and <j>(q,p) is transformed into a function of the
a r and 6|! constants, say (a v a 2 , b l9 6 2 , ).
All this is briefly and picturesquely expressed by saying that
any path " lies on " some energy -hypersurf ace. If the
energy is e, then, of course,
<A K, a 2 , b l9 6 a , ) =- e . (16.3.2)
Now, in general, this equation can be satisfied by an
infinity of different values for the set of constants b l9 b 2)
, b 2 f J for (16 . 3 . 2) is only one equation in 2/ quan-
tities &x, 6 a > b 2 j. Thus on one energy-hypersurface
there is in general an infinity of undisturbed paths of the
particle. Of course it might happen that only one of the b r
constants appears in the function iff (a, b) y and if it appears
in a term of the first degree, there is only one path. Never-
theless, whether the paths are numerous or not, there is no
guarantee that they " fill up " completely the extension (of
dimensionality 2 / 1) of the hypersurf ace. Giving every
value from oo to + oo to t, and all the suitable values to
&!, 6 2 , b 2 f, we obtain an infinity of sets of values of
?i, > <?/>#!> > P/> satisfying
^ (?> P) = c,
158 STATISTICAL MECHANICS FOR STUDENTS
but that is no proof that we have thereby obtained all the
sets of values which satisfy this equation. This point should
not be overlooked. It has an important bearing on the
arguments which will be advanced later to justify the
postulates of statistical mechanics.
16.4 Quantisation of the Paths. Degeneracy. Still con-
fining ourselves to the internal motion of the molecule, we
know that any path* is characterised by the values of the
/ action-integrals which are derived in the analysis of the
motion into its latent periodic elements. These are, in fact,
functions of the constants, and, of course, will vary from
path to path. Also the energy can be expressed as a function
of these / action-integrals.
Quantisation of the paths is effected by selecting those
values of the action-integrals which are whole-number
multiples of h. This obviously selects certain energy -
hypersurfaccs as relevant in quantum atomic physics, the
others being irrelevant. The journey of the representative
point from one of the quantum hypersurfaces to another in
a quantum jump is an occurrence whose details are not
disclosed by any part of the theory of stationary states. It
is regarded as occupying a time so brief that only one
characteristic of it comes into the discussion, the amount of
energy emitted or absorbed. The contrast to classical theory
where the journey from one hypersurface to another during
a period of disturbance is supposed to agree with the laws
of dynamics, is marked. In quantum considerations, it is
the quantum hypersurfaces which are all important, and
the quantum paths on them. Interesting questions will
arise as to the number of quantum paths on one energy-
hypersurface. If in E(J l5 J 2 , , J^) we substitute
r-Ji, rji, ,r f h for the several J r , we obtain a function
c(r l9 r 2 , TJ) of the quantum-numbers of the path,
which is equal to the energy corresponding to the hyper-
surface on which it lies. Now if the r g variables were
* The internal motion is naturally visualised as the motion of several
submolecular particles (atoms, electrons) in orbits ; but do not confuse
" path " with any particular orbit. The path is a synthesis not only^of the
geometric features of these orbits, but also of the actual conditions of
movement of the sub -particles in them.
DISTRIBUTION OF A SYSTEM IN ENERGY 159
capable of taking any values, then in general many quantum
paths would lie on a hypersurface, but as they are integers,
we cannot maintain this statement ; for we cannot assert
that in general an equation like
can have a solution in integers at all for a given value of c,
and if it has one such solution, we cannot therefore assert
that it has any more such solutions than this one. Still,
there are certain easily-defined cases in which there are more
quantum paths than one on any of the quantum hyper-
surfaces. Thus, if the energy function E(J) happens to be
a function of 3i + J 2 , J 3 , J 4 ...... , J/, then e(r) is a
function of r l -\~ r 2 , r 3 , r 4 , ...... r f . Thus any pair of
integral values of r a and r 2 which have the same sum will
yield the same energy. But different values of r l and r 2 ,
even if they have the same sum, when combined with one
set of values for r 3 , r 4 , ...... , r f , correspond to different
quantum paths which are therefore on the same energy -
hypersurface. It will be recalled that the fundamental
frequencies of the internal motions in the molecules are given
by
3E(J)
v,
so the condition of affairs just mentioned means that v l = i> 2 ,
or two of the fundamental periods are the same. In reality,
E(J) is a function of only / 1 variables, since the single
variable, J, + J 2 , takes the place of J l and J 2 . We cannot
enter into the dynamical theory of this matter here, but the
reader will probably not find much difficulty in accepting
the statement that the motion appears to be one in which
there are not /, but only / 1, degrees of freedom. It is
said to be " degenerate " on that account ; the question is
one of importance in statistical theory, as we shall see
presently. As a matter of fact, any linear relation between
the frequencies, such as
a i v i+*2 v 2 + + ajv f = Q . . (16.4.1)
160 STATISTICAL MECHANICS FOR STUDENTS
where the a f are positive or negative integers, produces
degeneracy.* The internal motion in the molecules appears
to be deprived of one of its internal degrees of freedom ; but
what is more germane to our purpose is that there are in
such cases a number of different quantum paths on the same
energy-hypersurface. If two relations, such as (16 . 4 . 1) hold,
the system is doubly degenerate ; the molecule appears to
have lost two of its internal degrees of freedom, and the
possibility of multiplicity of paths on the same hyper-
surface is increased.
We saw that in statistical-mechanical theory the quantum
paths take the place of equal-sized phase-cells in counting
complexions. Thus in the most probable state of the system
of molecules the number of molecules which are in the
quantum-state denoted by the quantum -numbers r l9 r 2 ,
...... , ry, is proportional to
If now there are w 8 quantum paths on the energy -hyper-
surf ace corresponding to an energy e g of the molecule, then
in the most probable state of the system the number of
molecules having the energy 8 is proportional to
If classical conditions held for the internal motion, it was
pointed out that in the most probable state the number of
molecules having a definite energy, would be really zero,
i.e., statistically. A statistical statement could only be
made concerning molecules whose energies lie between and
c + Se ; the number is
- C er^ x(<0 Sc
The corresponding full statement for the quantum con-
ditions is that the number of molecules which have an
* See the Appendix to the Chapter.
DISTRIBUTION OP A SYSTEM IN ENERGY 161
assigned energy e consistent with one or more stationary
states is
C w er*< h f
where w is the number of states having this energy. The
expression wh* replaces x(<0 Se. Both have of course the
same physical dimensions, viz., those of the / th power of
action.
APPENDIX TO CHAPTER XVI
IT will be less troublesome to follow the discussion on
degeneracy if we consider the condition (16.4. 1) for three
degrees of freedom. The reader can easily make the exten-
sion to any number. In that case there are three action-
integrals, and the condition
04 V-L + a 2 v 2 + a s V 3 ~
where 04, a 2 , a 3 , are three integers, positive or negative,
implies that
8E(J) 8E(J) 8E(J) __
a l -X-f + a 2 --7 + a 3 -3-7 V-
Oj l C J 2 (7 J 3
Now introduce the substitution
I 1 ai Jj -f- 12 J 2 ~f~ ^13 JB
"2 == a 21 ^1 "f" a 22 ^2 ~t" ^23 ^3
I 3 = a 31 eTj + a 32 J 2 + a 33 J 3
where the a rs are integers. We can solve for the J r in terms
of the I r and substitute in E(J) ; the result is a function of
Ij_, I 2 , I 3 , say F(I). It is easy to see that
8E(J) = 8F(I) 3Ii 8F(I) 8^ 2 8F(I) 8I_ 3
8J X " 8Ii *8Ji 8I a "8Ji 8I 3 '8Ji
8F(I) 8F(I)
and similarly
3E(J) __ 8F(I) 8F(I) 8F(I)
8E(J) 8F(I) , 8F(I) , 8F(I)
-^r~- == a !3 -3T~" + ^23 -~f- + a 33 ~-^f
dJ 3 oL l ol 2 o! 3
8.M.
162 STATISTICAL MECHANICS FOR STUDENTS
So that the condition for degeneracy, is satisfied if
011 <*1 + 012 a 2 + 013 a 3 =
a 2l a 1 + 022 a 2 + 023 a 3
a 31 04 + a 32 a 2 + a 33 a 3 = 0.
Now if we choose any two integers m and n, we can find
another number p such that
If p is not an integer, it is nevertheless a commensurable
number since a v a 2 , a 3 are integers ; hence, by clearing of
fractions we find three integers b ly & 2 , ^3> suc h ^hat
b 1 a 1 + 6 2 a 2 + 6 3 a 3 = 0,
and from the method of finding them it would appear that
there exists an unlimited number of ways of doing so. Thus
the three conditions for degeneracy written above are
perfectly feasible, but they would appear to be only three
of an unlimited number of such conditions. As a matter
of fact there is really one too many. It is a well-known
theorem in simultaneous equations that we cannot satisfy
three equations
a n x + a l2 y + a 13 z =
021 X + a 22 V + 23 * =
31 X + a 32 y + 33 Z =
simultaneously for any arbitrary values of the nine co-
efficients. If they are to be true simultaneously, then it
must also be true that
3i = A a n + /A a 2l
32 = A 012 + p 022
a 33 = A a 13 + V* 023
where A and /z are some two multipliers. Thus it appears that
J s = A Ii + p I 2 ,
and thus we have reduced E(J X , J 2 , J 3 ) to a function of
two variables, I and I 2 , since I 3 is a linear function of these
two. The mechanical system then behaves as if it had only
two independent action-integrals, two independent periods,
and only two degrees of freedom.
When it comes to quantisation, we see that the energy,
DISTRIBUTION OF A SYSTEM IN ENERGY 163
being expressible as a function of I x and I 2 , is expressible
also as a function i/t(p, a) of two integers p and a, which are
given by
P = 11 r i + 12 **2 + 013 7*3
a == a 21 r x + a 22 r 2 + a 23 r 3
where r l9 r 2 , r 3 , are the quantum numbers of a path. Clearly
this is compatible with the same energy for different sets of
values for r l9 r 2 , r 3 . Thus the quantum numbers r^ + a l9
r z + a 2> r s + a s> would give the same p and or, or for that
matter, r^ + ka l9 r 2 + fca 2 , r 3 + ^3, would do so likewise
where k is any integer. Still the choice is not unlimited ;
for we must bear in mind that the quantum numbers must
be positive integers, while a l9 a 2 , a 3 are not all positive.
This condition will, of course, limit the choice of suitable
values of k.
It may be of interest to point out the general sort of
evidence which experiment can give as to existence' of
degeneracy in atomic systems. The most notable example
is the Zeemann effect. A spectral line of an atom is split
by a magnetic field into a number of lines with slightly
different wavelengths. This means that the mechanism in
the atom has several different elements of periodicity which
have the same frequency. The magnetic field affects each
element rather differently, and the alteration of frequency
produced is not quite the same in each case, and so each
element gives unambiguous evidence of its presence.
CHAPTER XVII
QUANTUM THEORY OF THE SPECIFIC HEATS OF GASES
17.1 Specific Heat and the Internal Oscillations of the
Molecules. From the considerations outlined in the previous
three chapters, we find that the internal motions of the
molecules contribute to the energy-content of the system
C 27 , w 8 e-^*
where the summation is over all the energy hypersurfaces,
and the constant C is determined by the equation
C Sw s e~^* = n.
Let us represent the lowest quantum energy state by
the suffix zero, and put C e~^ = C . By Bohr's second
postulate, if a quantum jump takes place from the hyper-
surface 8 to the lowest quantum state, the frequency of the
radiation emitted is i> 8 where
, Co= hv 8 .
Thus
C (1 + 27 w 8 e-^"') = n,
8 = 1
and the energy-content is
C, (e. + 27 c. w 8 e- hv <).
=i
Combining these, we see that the energy-content is
n . e + 2 c. w s e- h ">
I +Zw t *->*> ' ' ' I"- 1 ' 1 *
and if we differentiate this expression with respect to 0, we
obtain the contribution made by the internal motions of the
molecule to the specific heat of the system of molecules. If
we do so, we obtain an expression whose numerator is
nh(Zv 8 w 8 e-* hv (1 + 27 w e-^'w
164
SPECIFIC HEATS OF GASES 165
and whose denominator is
(1 + 2 w, e-'*"') 2 .
The latter is positive and greater than unity, so the con-
tribution to the specific heat turns out to be less than
nhjkd 2 multiplied by the sum of a number of terms such as
(v e A- v v e v } w w e~^ v 8+ v u)
\ V 8 e # I V U *U V U t S V 8 e / w W U ^
This latter expression is equal to
Thus the contribution is less than
/ __ \ 2
vi z* y y i s u i ID in 0~t jL ' i ( i '8^~ i 'ui
nk ^^(~j: r ) w > w ^
Now the frequencies involved are according to spectro-
scopic evidence in the ultra-violet, luminous or high infra-
red ranges of the spectrum ; their order of magnitude is
certainly not lower than 10 14 . That being so, it will be easily
seen that for ordinary temperatures the value of the index
in any exponential factor is numerically greater than 15,
and in the majority very much greater. But e~ 15 and other
experimental factors still smaller, will practically render the
multiplier of nk in the previous expression negligible. The
smallness of the contribution is still more marked for low
temperatures. Only for very high temperatures would the
internal degrees of freedom begin to contribute an appre-
ciable fraction of nk or R for each degree of freedom in the
internal structure of the molecule. In this way the quantum
theory surmounts one of the difficulties of the older theory,
which required for each degree of freedom a contribution to
the specific heat of R/2 as regards kinetic energy and an
amount of the same order of magnitude as regards potential
energy. In general terms, and putting aside the rather
clumsy expressions involved in the analysis, the spectro-
scopic evidence shows that the great majority of the mole-
cules will be in the lowest internal quantum state, by reason
of the smallness of the factors, such as e~^ s . Thus the
166 STATISTICAL MECHANICS FOR STUDENTS
internal energy will not differ much from n . At a some-
what higher temperature there will be a very small change
in the distribution of the molecules among the higher
quantum states ; the exponential terms are not seriously
affected ; practically the internal energy will still be H O , and
the quotient of the change by the rise of temperature is
insignificant. Only at extremely high temperatures when kQ
would be approaching in value to some of the hv 8 , would the
change in internal energy with rising temperature begin to
show itself in the specific heat.
17.2 Molecular Rotation and the Specific Heat of a
Diatomic Gas. The order of magnitude of the frequencies
involved in electron motions within the atoms or even in the
relative vibrations of the atoms constituting a molecule is
therefore so great that no contribution to the specific heat
of a gas can be expected from this quarter except at extremely
high temperatures. Yet we know that the ordinary trans-
latory movements of the gas molecules can only contribute
an amount 3R/2 to the thermal capacity of a body of gas,
and this is too small under ordinary circumstances except
for monatomic gases. We saw in Chapter V. that in
the case of diatomic gases the balance, approximately R,
could be accounted for reasonably enough by assuming a
further two degrees of freedom involved in a rotation of
the m'olecule about an axis at right angles to the line joining
the centres of the two atoms ; rotation about this latter line
itself is excluded from consideration for reasons stated in
that chapter. Nevertheless experiments, especially on
hydrogen, show that this contribution does not remain even
approximately near R as the temperature decreases ; the
evidence is very much in favour of the view that as the
temperature approaches absolute zero, the contribution of
molecular rotation to the specific heat diminishes asymp-
totically to zero. This suggests at once that in some way
the rotational motion is quantised and not subject to classical
equipartition, while the frequencies involved are of such
magnitude that the average energy must be close to k6 for
ordinary temperatures, but much less as 6 decreases below
normal values. Now the average energy is
SPECIFIC HEATS OF GASES 167
hv
exp (hv/kO) 1
or 7 /j x
e*~^T
where x = hvjkO. Putting 9 = 300, k = 1-37 X 10"" 18 and
h = 6-55 X 10~ 27 , we see that to make x = 1, v must be of
the order 6 X 10 12 . Consequently frequencies between 10 12
and 10 13 would, at ordinary temperatures, approximately
involve the classical partition of energy since for such condi-
tions xj(e x 1) would approximate to unity. But at much
lower temperatures x would increase to the order of magnitude
of 10 or even 100, and then x/(e x 1) is practically negligible.
Now, as a matter of fact, it has been discovered that in the
absorption spectrum of water-vapour there actually exist
some lines whose frequency are in the region about 5 X 10 12 .
These lines are usually referred to as the rotation-spectrum ;
and as a piece of additional evidence there has also been
observed a well-marked effect produced by rotations of such
frequencies on the lines in the infra-red of order 10 14 , which
arise from the atomic vibrations in the molecules of water-
vapour and the hydrogen halides.
To be sure we are not dealing here with an oscillator in the
Planck sense, having one frequency for all amplitudes, and
the treatment just outlined is too inadequate. But a
method of quantising a rotation, easily deduced from
the considerations of Chapter XV., forms the basis of a
theory which accounts for the behaviour of diatomic gases
as regards their specific heats, apart from some minor
discrepancies, which appear to arise rather from our lack of
knowledge of the actual structure of the molecules than from
any serious deficiency in the theory.
A diatomic molecule pictured as a dumb-bell has the line
joining the atoms as one principal axis of inertia. Any two
lines at right angles to this and to each other and passing
through the centre of gravity can be taken as two other
principal axes of rotation. The atom is a " symmetrical
top/' but will the reader carefully bear in mind that rotation
about the axis of symmetry, i.e., the line joining the atoms,
168 STATISTICAL MECHANICS FOR STUDENTS
does not come into consideration ? One reason for excluding
this rotation was given earlier in the classical treatment ;
another reason, more in keeping with quantum ideas, will
emerge presently. It is with rotations about axes at right
angles to the line of symmetry that we are concerned. Let
A be the symbol for the moment of inertia of the atom about
such an axis. This is as a matter of fact equal to M l r x 2 +
M 2 r 2 2 where M x and M 2 are the masses of the atoms, and
r l and r 2 are their distances from the centre of gravity, and
it is not difficult to show that this is also equal to M a 2 where
a is the distance separating the atom-centres and
JL--L + 2-.
M M 1 M 2
The angular momentum round the axis is Aw if co is the
angular velocity. The action accumulated in one period,
i.e., in one rotation is then 2 TT A o>, and so, according to
quantum views, rotations with any angular velocity cannot
exist for an interval of undisturbed rotation, but only
rotations for which the action-integral 2 n A co has a value
such as hy 2 h, 3 h, etc.* Hence the " rotator " can only
have one of a series of discrete angular velocities co 1? o> 2 ,
cu, ...... where
When rotating with one of these velocities it is in a quan-
tum state defined for the moment by one quantum number r.
But one number is really insufficient. We have two degrees
of freedom to deal with (the third, rotation around the axis
of symmetry, has been definitely excluded), and we must
have two quantum numbers. We have here, in fact, an
example of degeneracy, and in the early days of the theory
this feature of the situation caused some trouble for a reason
which will appear presently. Leaving this ambiguity on one
side for a moment, we see that the energies of the quantum
* The reader will observe that angular momentum being the product
of mass, distance squared and angular velocity actually has the same
physical dimensions as action.
SPECIFIC HEATS OF GASES 169
states of the rotator defined by (17. 2. 1) are given by
equations such as
m <17 - 2 ' 2)
Thus the energy in a quantum state varies as the square of
the quantum number, a rule quite distinct from the rule
which holds for a Planck oscillator ; in that case the variation
is with the first power. We would now be in a position to
obtain the average energy of a molecular rotator in the
system if we knew the number of quantum states consistent
with this energy, for as we have seen, it is equal to
2 w
r
2 w
(17.2.3)
One of the earliest attempts to use this expression was
based on the assumption that the w r are each unity and that
the result thus obtained should be doubled so as to take
account of the two degrees of freedom really involved in
rotation about an axis at right angles to the symmetry-line
of the molecule. It will be instructive to carry out this
evaluation before proceeding to later attempts to cope with
the difficulty. On such a view the rotational energy of the
system is
00
2 e e'^r
2w l
d fJL
fj (
2n log 1
,
where
a =
170 STATISTICAL MECHANICS FOR STUDENTS
The rotational part of the thermal capacity of the n mole-
cules of gas is then equal to
dO dfju
5 l^rr F /,-of*
At low temperatures //,, and therefore a, are relatively large.
The series S e~ ar * practically reduces to 1 + e~ a , and the
logarithm of this to e~ a ; thus the rotational thermal
capacity is equal to
,72
-
and since for large values of /x or a, the factor e~~ a " swamps "
ju, 2 , this has a limit zero as /z increases indefinitely. For high
temperatures a is small and since
1 + e~ a + e~ 4a + e- 9a + ......
= a -l{ Xl + e -V (x 2 -xj+e -*>\xz - x 2 ) + e~*>*
where x 1 a*, x 2 = 2a*, x 3 3a J , etc., the series S e~ ar *
can be approximately written as a definite integral
f 00
a~M e~ x * dx,
Jo
which is ^ (?r/a)*. Thus at high temperatures the rotational
thermal capacity has as its limit
which is equal to
The conclusions concerning the extreme limits are quite
satisfactory ; but, unfortunately, a closer investigation shows
SPECIFIC HEATS OP GASES 171
that the thermal capacity rises to a maximum above R and
sinks to a minimum below it before reaching the limit R.
The experimental observations do not bear out this con-
clusion. Various other suggestions were made for dealing
with the doubtful situation arising from the degeneracy of
the motion, but all suffer more or less from the defect
mentioned above, viz., a tendency for the computed value
to rise somewhat above the experimental value. Still they
are an improvement on the earliest result obtained by the
mere doubling method. The way in which the degeneracy
of the motion is removed involves a rather wider knowledge
of dynamical science than is assumed in this book. It must
suffice to say that one plausible suggestion leads to the
conclusion that there are really r different quantum states
with the energy r 2 k 2 /S n 2 A , and so w r should be put equal
to r. Incidentally, this implies that in no state is the energy
of rotation absolutely zero, the lowest quantum state has
the energy of rotation h 2 /S7T 2 A. This feature, viz., that
the lowest quantum state is not one devoid of rotational
energy, is a common feature of all the hypotheses used to
explain not only the thermal behaviour but also the results
of spectroscopic analysis. In the same manner as we pro-
ceeded above, we now find that the rotational thermal
capacity of n molecules is
72 CO
R jLt 2 - log 27 (r e~ ar *)
At low temperatures this vanishes in the limit and at high
temperatures the series approaches the value of the definite
integral
/<#
x * dx,
_ f
a" 1 ! x
Jo
i.e., l/2a, which gives us a value R for the above expres-
sion at the upper limit of 0. But computation of the
series still shows a discrepancy at moderate values. Thus
about 200A the computed value for hydrogen is -82 R
as against the experimental -72 R, a rather serious differ-
ence in view of the claims made by the experimentalists
as regards the precision of their measurements. By the
172 STATISTICAL MECHANICS FOR STUDENTS
time, however, we reach the temperatures of our normal
surroundings, the discrepancy disappears, the values at
freezing point of water being -936 R and '937 R respectively.
The reader may recall the statement made earlier that
recent spectroscopic work calls for " half -quantum numbers "
in certain cases, meaning that although the quantum
numbers increase by integral amounts from quantum state
to quantum state, they are not necessarily integers them-
selves, but may have values such as '5, 1-5, 2-5, etc. This
suggestion has been tried in this problem, and r being treated
this way, it is found that w r can either have the series of
values 2, 4, 6, 8, or 1 , 3, 5, 7, In either case
this method gives better agreement with experimental facts
than those based on integral values of the quantum numbers.
As mentioned above, the source of the discrepancies may lie
in the assumption of a constant A throughout ; at higher
temperatures there may be a " stretching " of the molecules
owing to increased rotation and some change in the moment
of inertia. The computations incidentally enable us to find
a numerical value for A ; the values differ somewhat
according to the method used, but they all agree in giving
something of the order of magnitude 10~ 41 in C.G.S. units.
This result is of the same order of magnitude as the values
obtained from the observation of the rotation spectra and
rotation-vibration spectra of vapours of water and the
hydrogen halides, and is consistent with the value 10~ 24
gram for the mass of the hydrogen atom and -5 X 10~ 8
cm. for the diameter of the hydrogen atom. The latter
are derived from the kinetic theory of gases or from Bohr's
theory of the Balmer series, and it is assumed that the atoms
in the molecule are close together.
We can now give the very simple explanation on quantum
lines why we leave out of consideration rotation round the
axis of symmetry of the molecule. The moment of inertia
of the molecule round this axis is much smaller than A . All
our present knowledge of atomic structure points to the
conclusion that the mass of an atom is practically con-
centrated in the small nucleus, whose linear dimensions are
of a much smaller order of magnitude than the radius of the
SPECIFIC HEATS OF GASES 173
atom itself, i.e., the radius of the outer electron orbits in the
normal state of the atom. This latter radius is the order of
magnitude of the distances of the atom centres from the
centre of gravity which are used in calculating A ; but in
calculating the moment of inertia around the axis of sym-
metry we would use distances of an order of magnitude
given by the size of the nucleus. Call this latter moment of
inertia C. If rotation about this axis came into play, the
smallest possible angular velocity would be given by h/2irC
and the corresponding energy by h 2 /$7r 2 C. In comparison
with h 2 l87T 2 A, this would be very large ; the chance, there-
fore of a molecule being in the lowest quantum state of
rotation around the axis of symmetry would be very small
relative to the chance of it being in the lowest, or even in
many a higher quantum state of rotation around axes at
right angles to the symmetrical line. Molecules rotating
round the axis of symmetry are in consequence too few in
number to affect the final result.
CHAPTER XVIII
THE ELASTIC SPECTRUM OF A LINEAR LATTICE
OF COHERING PARTICLES
18 . 1 The "Co-ordination " of a Chain of Particles. For
a real understanding of the manner in which the quantum
hypothesis can be applied in the treatment of the specific
heats of solid bodies, it is necessary to know something of the
mathematical method by means of which the analysis of the
irregular heat motions of the atoms into component simple
vibrations is effected. The reader is probably aware that
the physical and chemical facts concerning crystalline
materials support the view that a " molecule " is apt to lose
its identity in a solid. The constituent atoms of the mole-
cules are arranged in space lattices, corresponding atoms of
each molecule forming one lattice, other corresponding
atoms forming another lattice, the lattices interpenetrating
one another. Under such an arrangement it is hardly
possible to say that one particular atom is associated
with another particular atom to form a molecule. The
partnerships set up when the substance is liquefied or
vaporised hardly exist in the solid state. We shall, therefore,
take as an ideal simple solid a group of particles of one kind
arranged in a cubical lattice. This will form a convenient
model for a monatomic solid.
In presenting the mathematical method referred to, it
will be advisable to show its application in the first instance
to something even simpler than the ideal solid, viz., a row
of particles cohering together by reason of strong attractions
to form a chain, which, however, is not really rigid, but is
capable of oscillating and displaying an enormous number of
forms following one another in a manner determined by
dynamical laws.
Every one with a musical ear can detect in the sound of a
174
ELASTIC SPECTRUM OF A LINEAR LATTICE 175
note played on any instrument a series of simple notes with
definitely related pitches, the so-called fundamental tone
and its overtones. It is well known that these sound sensa-
tions are related to vibrations in the sounding body which
can be analysed into more elementary vibrations each
having a definite frequency. One of the most customary
ways of illustrating this fact in works on Acoustics is to take
the violin or piano string as an example. To apply the
mathematical method, in its most simple form, we idealise
these and conceive a string without any physical property,
except length, mass and the capacity to support a tension
and a slight stretching without breaking. It is in the first
instance regarded as a linear continuum, so that the smallest
fraction of the distance between its ends contains its pro-
portional amount of mass. When such a string vibrates, we
can imagine that its form at any instant can be rendered
permanent for a while, so that we can study it at leisure.
A cinematograph film of its behaviour, for example, can be
stopped with a particular picture showing. The possible
shapes have an infinite variety and complexity, yet they
can be dealt with in a very powerful and elegant manner by
a famous theorem due to Fourier. The geometry looks
absolutely unmanageable, but the analysis is not at all
difficult to understand. Let us assume that the string is
fastened at its two ends. Distance from one of these ends
along the string direction we shall denote by the symbol #,
the whole length being /. Let us also for convenience assume
that the vibrations are confined to one plane, i.e., that each
element of the string has only one degree of freedom. The
displacement of an element in this plane from its equilibrium
position, and, of course, at right angles to the direction of the
string, we shall denote by . The fact that the string has a
definite shape at all at our moment of observation is expressed
by saying that is a function of x. In fact, when we write
we mean this. We have written down the " equation of the
curve." Yet if we should glance at any of the shapes of
vibrating strings that have been actually obtained with
176 STATISTICAL MECHANICS FOR STUDENTS
violins, etc., we might well be dubious about the possibility
of finding the actual " form " of the function *jj (x). Yet
there is a way of doing it which enables us to obtain i/j (x)
approximately with the possibility of carrying the approxi-
mation to any degree of accuracy necessary. Formal proof
cannot be given here, but reference to any standard text on
the calculus will substantiate the following result
. . x .770;. . ZTTX , , . TTTX
yt (x) = q l sin + q 2 sin + + q r sin
V V L
+ adinf (18.1.1)
where the coefficients are definite integrals defined thus :
TTTX ,
2 ( l . . . .
ft = y <A ( x ) sm
I JO
If the shape were actually drawn on a sheet of paper, we
could, by multiplying each ordinate by the value of sin
r TT x/l at the point, obtain a new curve whose area would
be the coefficient q r multiplied by 1/2. Work of a kind
analogous to this is actually carried out to-day for
practical ends, e.g., in analysis of the tides. The reader will
therefore realise that the statement that we could find the
equation of the curve to any desired accuracy is not merely
" theoretical; " it is quite practicable, if somewhat tedious
at times. The salvation of the practicability lies in the fact
that in a great many cases the series is so highly convergent
that a half-dozen terms or even fewer serve very well. (In
the most highly developed " harmonic analysis " of the
tides at the present day, investigators seldom use more than
two dozen harmonic constituents. ) Even such an apparently
intractable shape as a " zig-zag " is extremely well repre-
sented by six or seven terms.
So far this is a matter of geometry and analysis. Now we
take up the kinematic side of the matter. The string does
not stay in this shape. Shape after shape follow in a con-
tinuous succession depending on the tightness of the string
and the shape from which we " let go." In the analysis this
succession of shapes corresponds to a succession of values for
the group of coefficients, q r . Each of these is in fact a function
ELASTIC SPECTRUM OF A LINEAR LATTICE 177
of time. If we knew what these functions were, we would
have in that knowledge summarised the whole history of the
string's behaviour. In short, #!, # 2 > ...... are " co-ordinates
of the string." Here we take a more general attitude towards
the word, co-ordinate, than hitherto. It is not to be merely
restricted to Cartesian or polar methods of specifying the con-
figuration of a system. This will, in fact, be a useful illustration
to fall back on when we come to treat the most general way
of stating the laws of dynamics in a later chapter. Any set
of quantities which when known specify precisely the com-
plete configuration of a system can be regarded as co-
ordinates of the system. The very name " arranged to-
gether," suggests this.
If now we wish to find the functional forms which show
how the various q r depend on t, we must naturally use the
laws of dynamics. It is known from the application of these
that the displacement is a function of x and t which
satisfies the partial differential equation
where T is the tension of the string, and M is the mass of
unit length of it. From this and equation (18 . 1 . 1) it follows
that
7r 2 T * . TTTX d\ . TTTX
-- 27 r 2 q r sin - M sin -
I 2 r -i I r-i dt 2 I
If this is to be true for any value of x, it follows that each
q r satisfies an equation of the type
d\ 2 7T 2 T
_ L 7*2 ___ g
dt* I 2 M ^
Hence
$ = a r sin (w r t e r ) . . (18.1.2)
where
and a r and e r are arbitrary constants. The constants o> r are,
of course, independent of the initial conditions ; they are
178 STATISTICAL MECHANICS FOR STUDENTS
determined entirely by the nature of the system. The o> f
thus form a harmonic series of pulsations (pulsation is 2 TT X
frequency) whose fundamental is (rrjl) . (TjM)^. On the
other hand, the a r and the r are the usual integration con-
stants entering into the solution in the integration of the
equations of motion. They are arbitrary in the sense that
if we start the string to vibrate by releasing it from one
configuration, and then give it another vibration with a
different initial form, the two sets of a f and r will be
different. A glance at (18 . 1 . 2) will justify the use of the
word " amplitude " for the a n but again in a more general
sense than hitherto, not being confined to the amount of
excursion from side to side which any element of the string
makes. To make this clear, let us suppose that we choose all
the a r to be zero except a l ; all the q r are also zero except q l9
and we find for the displacement at time t of an element of
the string situated at x the result
. . . . TTX
a l sm (a) l t ej sm .
i
If x = or I, this is zero, as it must be since the ends are
fixed ; if x = 1/2, the vibration is a l sin (coj t j), and so
a l is the amplitude, in the ordinary sense, of the vibration of
the middle element of the string ; but for any other position
this is not so ; the amplitude, in tlie customary sense, is
a l sin (TT x/l), which gradually decreases to zero as we
approach either end. The string is then vibrating as a whole
between two extreme sine forms
y . TTO;
=a 1 sm
, y . TTX
and = a 1 sm .
L
This is its fundamental vibration. If, however, we make
all the a r zero except a 2 , the displacement is given by
v i \ %1TX
= a 2 sm (co 2 t e 2 ) sm .
i
This vanishes always at the middle point as well as at the
ends ; there is a " node " there. Only at the points of
ELASTIC SPECTRUM OF A LINEAR LATTICE 179
quadrisection will the amplitude of displacement be a 2 >
there are two " antinodes " situated at those points. The
string vibrates as it were in two halves between the extreme
form
j. . %7TX
4 = a 2 sin
I
and the extreme form
2-7TX
= a 2 sin
I
which are two complete sine-curves. The extension of this
is obvious. The a r are the amplitudes of the q r co-ordinates.
In the most complicated vibration of the string there are
latent all the simple vibrations with an ascending series of
harmonically related frequencies.
So far we have been dealing with the ideal string familiar
in works on Acoustics. We must now point out how the
result is modified if we replace the string by a chain of
equally -spaced particles. For one thing we cannot be
involved in an infinite series. If there are / particles, each
with one degree of freedom, we only require / co-ordinates,
no matter how chosen, to specify any configuration. So that
gives us a broad hint that instead of an infinite series we must
have something like this for an instantaneous form of the
chain
u, . . 7T X k . . . 2 7T X,. . , / flTXlf
* = <i sin * + < 2 sin j- + + ^ sin J ^
where x k is the distance of the k ih particle from one end and
j. its displacement. There is a distance l/(f + 1 ) between
each particle, assuming that the two end ones are anchored
to two immovable particles which we can suppose to be
labelled and / + 1 respectively. Thus the previous
equation can be written
v , krr . . . 2Ic7T . . , . fkrr
t = #i sm jq7j +^2 sm j^-j + + ^/ Sin jq7j
(18.1.4)
(Incidentally this makes and g f+ l zero as it should.)
Now investigation shows that this surmise is quite correct,
N2
180 STATISTICAL MECHANICS FOR STUDENTS
and dynamical analysis * leads to a functional dependence
of the/ co-ordinates, <f> v ^ 2 , ...... <f> f on t given by
</> r = a r sin (K T t r ),
where the a r and r are 2/ arbitrary constants of integration,
while the K r are / pulsations given by equations of the type
In this, T is still the tension, i.e., the attraction between two
neighbouring particles, m is the mass of one particle, i.e.,
M l/f, and I' is the distance between particles or l/(f +1).
The analogy between the <f> r and the q r or between the
to r and K r is obvious, and the question arises how far We can
make use of the mathematical methods, suitable for dealing
with a continuous medium, for the treatment of a medium
with a discrete structure not only in the case actually under
consideration, but also when three-dimensional bodies are
in question. In the first place
I'm Im
_/(/+!) T
_ .__.
Further, if r is small compared with /, and / is a very great
number
r 77 r TT
OlT-k ^^3
and _ r ?
KT ~~~T \M)
Thus, provided r is a reasonably small fraction of /, say
0-1 at most, there is practical equality between K T and to r .
But as we ascend into the higher ranges of the K T frequencies,
we cannot take them to be the ascending integral multiples
of the fundamental frequency. However, we can anticipate
that an application of the quantum theory will, in view
of the general feature, which has already been illustrated in
* See Routh's Rigid Dynamics, Vol. II., Chapter IX., or Rayleigh's
Theory of Sound, Vol. I., Chapter VI.
ELASTIC SPECTRUM OF A LINEAR LATTICE 181
other connections concerning the decreasing significance of
increasing frequency, turn out to produce results not far
from the truth if we overlook the departure of the higher
K T from the simple iile and assume that we are concerned
with / harmonic frequencies which are all integral multiples
of the frequency v given by
1
v = -
21 \M
That being so. we can plausibly assume that the <f> r agree
with the q r . The assumption is fully justified by a closer
analysis in the cases where the Fourier scries is convergent,
and it is only with such cases that we are physically con-
cerned.
Thus we have analysed the motion of any of the particles
in an undisturbed vibration of the chain into simple har-
monic constituents. If as before x stands for the distance
of a particle from one end
~ i . r TT x . IT 77 c t \
= S a r sm sin ^ e r j
where /T\*.
r V
\M)
It will be observed that the form of the chain exactly
repeats itself when t increases by 2 l/c. This is the period of
vibration ; it is also the time in which anything travelling
with a speed c would cover a distance equal to twice the
length of the string. Indeed it is easily inferred from the
general theory of the string that c or (T/M)* is the speed with
which a transverse pulse would travel along an unlimited
string with this tension and mass per unit length.
18.2 The Energy of the Chain. The velocity of the
particle whose distance from one end is x, is given by
V (i n m %
= Z q r sin
. r 77 x . , . .
= aj r a r sm sin (a) r t e r ).
182 STATISTICAL MECHANICS FOR STUDENTS
The kinetic energy is equal to - M I 2 dx, and since
1
. r 77 X . S 7T X j - ., .
sin sin dx = if r =f=
o I I
= _ if r
2
it is easy to see that the kinetic energy is
..... (18.2.1)
As regards the potential energy in any configuration, it is
the work required to produce the necessary stretching against
the tension T. If in this configuration an element dx of the
straight string is stretched to a length ds of an element of the
curve, the potential energy in that element is T (ds dx)
or T (ds/dx 1) dx. But
dx [ \dx
= f i /dty _i /<2\ 4 , 1
{ "*" 2 \dx) 8 \dx) J
Neglecting terms beyond the second power, since d^/dx is
small compared to unity, the potential energy of the string
turns out to be
1 T C l /d\*dx,
and since
^ _ ^ P r 7T a:
dx /r^i f Z
we can demonstrate as above that this energy is
7T 2 T *
2r*q* (18.2.2)
The kinetic energy and potential energy are by this
analysis each separated into / parts, any one part being
associated with a co-ordinate or its " velocity," where the
word " velocity " is now used to indicate the rate of change of
a generalised co-ordinate with respect to time and is not
ELASTIC SPECTRUM OF A LINEAR LATTICE 183
necessarily the actual rate of movement of any particle.
The similarity of the mathematical results to those for a
system of simple oscillators is obvious, and one important
feature is common to both, viz., the equality of the average
kinetic and potential energy of the linear lattice not only
in toto, but also in its analysed parts. The average kinetic
energy associated with the co-ordinate q l r is by (18 . 2 . 1)
Ml 2 2
"8 ^ ""
The average potential energy associated with q r is by
(18.2.2)
V * T 2 2
-*r r a -
and since o> r 2 = r 2 (n/l) 2 (T/M), the equality follows. Thus
from the point of view of mathematical procedure, the
atomic chain can be regarded as a complex molecule with
/ internal degrees of freedom, represented by co-ordinates
q r and velocities q r , each following the simple harmonic law ;
but it is necessary to be on guard against thinking that these
are each related respectively to one link in the chain. All
the q r and the q r enter into the displacement and velocity of
any individual atom of the chain.
18.3 The Statistics of a System of Atomic Chains. Having
thus dealt with the mechanical side of this problem as a
preliminary illustration for the solid lattice, let us turn to
the statistical aspect in a similar anticipatory vein.
Conceive that in a cubical enclosure an enormous number
n of such chains each containing / atoms are stretched across
from wall to wall, each anchored to the walls at its ends.
The enclosure also contains gas molecules, so that the whole
may be regarded as a mixture of gas molecules and " chain-
molecules." Exchange of energy goes on between the
various members of the system, and we can work out in the
usual manner the conditions of statistical equilibrium. Just
as we have generalised the meaning of co-ordinate and
velocity, so we can give a wider significance to momentum ;
the full import of the step will be more apparent in Chapter
XXIV, but we define the r th " generalised component of
184 STATISTICAL MECHANICS FOR STUDENTS
momentum " to be equal to the partial differential co-
efficient of the kinetic energy with respect to q r and denote
it by p r , so that in the present case
Ml .
Pr ~ 3r
We can now represent the 2f quantities q r , p r in a 2/-dimen-
sional phase-diagram or in/ two-dimensional phase-diagrams,
partitioning into phase-cells and counting the complexions
in any statistical state just as before. If the reader feels any
qualms about this procedure, feeling in a vague way that it
is hardly as justifiable a process as when we represented
actual co-ordinates and velocities or momenta of particles
in the usual sense in a phase-diagram, he can banish them
without any fear. He will see later that in so far as we can
accept the procedure to be a justifiable one in the latter case,
it is equally justifiable in the former. The particular
character of the laws of dynamics, when given their most
general mathematical form, takes care of that. Since the
energy involves only squares of co-ordinates and momenta,
we will arrive at the conclusion of equipartition of energy
on the average among the various co-ordinates and momenta
of the system, an amount | kd to each, so that the whole
system of n chains will, in an enclosure at temperature 6,
contain energy of amount nfkB, i.e.,fkO to each chain on the
average.
If, however, we adopt quantum views, we select the
quantum states of motion of a chain by giving various
integral values to the/ quantum numbers, s l9 s 2 , , <sy,
defined by
fp~ dq* == Si h
r i z i A
etc.,
where each integral is taken through a complete period of the
corresponding co-ordinate, i.e., 2 TT/O), for q r . These con-
ditions give in each quantum state of the chain definite
values to the amplitudes a x , a 2 > ay, so that for example
ELASTIC SPECTRUM OF A LINEAR LATTICE 185
the extreme forms between which a chain can swing to and
fro have not an " infinite variety " ; they form a discrete
series of shapes. The energy in the state s l9 s 2 > ...... > 5 />
is equal to
$i hv- -f- <9 2 hv% -f- ...... + Sj hvj,
where v r a) r /2 TT, and in the most probable statistical
state of the system of n chains the number which are in this
quantum state is given by
C exp { JJL h (s 1 v l + s 2 v 2 + ...... + s j "/)} (18.3. 1)
where the sum of all such expressions for all sets of values of
the integers (s l9 s 2 , ...... , s f ) is n. To find the average
energy associated with a particular co-ordinate, say q r , we
have to work out the sum
C llV T 2 S r exp { jLt h (8 l V l + S 2 l/ 2 + ...... +S f V f )\
(18.3.2)
and divide by the sum of the expressions (18 . 3 . 1). It is
easy to see that (18 . 3 . 2) is equal to
C (Es r hv r e-^Ar) [
where the E refers to summation of s r from to oo and the
Z' refers to summation with regard to the / 1 numbers
s v ...... ^r-i' s r -\- 1' ...... s f ver a ll possible values. The
summation of (18.3. 1) can also be represented by
C (E e'^'r) [E f exp {-ijih(s l v l + ...... + 8,^ v r ^
Hence the average energy sought for is
E s r Jiv r e~* 8rhv r
2 e-
and this, as before, turns out to be
Ay,
1 '
186 STATISTICAL MECHANICS FOR STUDENTS
On quantum views then the average energy of any chain
in the system of chains is
. . . (18.3.3).
r-l ef^r - 1
Since
we can write (18.3.3) as a definite integral regarding c/2l
as equivalent to a differential of frequency, Sv. Thus the
energy of the chain is
2 lr f hv ,
_ J ------ d v
CJ #* 1
where the upper limit is fc/2l.
18 . 4 A Superficial Lattice. A lattice of /particles arranged
in a square of side I with i particles in a row, and i rows
in the whole lattice (i 2 = /), can be treated in a very similar
way, by identifying it for practical quantum purposes with
its limit, viz., a square elastic sheet with a mass M per unit
area and a surface tension of T per unit length. The axes
OX and OY lie along two sides of the sheet, and the sheet is
supposed to be fixed along its boundary. It is known that
if is the displacement of a point (x, y) of the sheet, it is a
function of x, y and t which satisfies the differential equation
T
where c 2 = - .
M
From this we find a solution which satisfies the boundary
condition, that = if x = or I or if y or I. It is
* * 77 x . s TT y
sin*
i I
where the q r8 are i 2 (or/) " co-ordinates " following a harmonic
law of variation with time
q rs = a n sin (o> w < - e,,).
ELASTIC SPECTRUM OF A LINEAR LATTICE 187
The a rs (amplitudes) and r8 (epoch-angles) are arbitrary,
but the natural pulsations of the lattice are given by
as can be easily verified from (18.4.1).
All the necessary details for quantisation can be easily
supplied by the reader from the previous section, leading to
the average energy of any atomic sheet (regarded as a com-
plex molecule) belonging to a system of such sheets im*
mersed in an atmosphere of gas molecules. It is
(18.4.2)
r-n-i e^r* - 1
s\
where v n = (r 2 + s 2 )*.
To convert this into a definite integral in the same fashion
as before, we observe that all the natural frequencies of a
lattice which are not greater than a definite frequency v,
are the same as the number of pairs of positive integers the
sum of whose squares is not greater than (2/*>/c) 2 . If the
integers are plotted on squared paper ruled in unit lengths,
we can realise that this number is the number of units of
area in a quadrant of a circle drawn on this paper having a
radius 2lvjc ; so it is
It follows that the number of natural frequencies of a
lattice which lie in an elementary range v to v -f- 8v is equal
to
Sv .... (18.4.3)
Thus to arrive at a definite integral which replaces (18.4.2)
we must collect all the terms which correspond to v + $v >
v r > v \ these will supply one element to the integral ;
their individual value is taken as liv\(e^ v 1), and the
188 STATISTICAL MECHANICS FOR STUDENTS
number of them is (18 . 4 . 3). So the energy of a lattice on
the average is
dv (18.4.4)
where v f = t (i 2 + i 2 ) 1
2i i
e? hv 1
c
CHAPTER XIX
THE ELASTIC SPECTRUM OF A CUBICAL LATTICE
19.1 Some Preliminary Mathematical Statements.
I. The wave equation.
If f(x, y, z, t) is a function of four variables which satisfies
the partial differential equation
S 2f d 2f d 2f ^ l d 2f
Sx* dy* dz* c* dt*'
it is called a " wave function." It can be easily verified that
any function of the expression ax + fty + yz + f]t is a wave
function if the constants a, /?, y, 77, satisfy the relation
a 2 +/? 2 +y 2 -^ . . . (19.1.1)
The reason for the name is fairly obvious in this special case.
If x, y, z, t, represent space and time co-ordinates, the value
of the function at a place x l9 y^ z l and time t l is the same as
its value at the place x 2 , ?/ 2 , z 2 and time t 2 provided
ax l + fiiji + yz l + r]t L = ax 2 + /fy 2 + yZ 2 + rjt 2 .
In short, the value which the f unctign has on any plane
ax + py + yz + 7]t l
at time t 1 was or will be the value on the plane
ax + fly + yz + ^ 2 -
at time t 2 . These planes are parallel to one another, and
separated by a normal distance
i.e., according to (19 . 1 . 1) by the distance
c ( ~~ t).
189
190 STATISTICAL MECHANICS FOR STUDENTS
This clearly can be taken to refer to the propagation of any
value of the function in a direction defined by the direction
cosines a/ (a 2 + /? 2 + y 2 )*> etc., with a velocity c. The con-
stancy of the value of the function over a whole plane at
one instant is indicated by the use of the term " plane
waves. " As the wave -equation involves only first powers
of the differential coefficients, the sum of any number of
particular solutions is also a solution ; so the propagation
of any quantity by a group of plane waves in any number and
in any directions would still yield at each point and instant,
a quantity satisfying the wave equation.
An important special case of this occurs when the function
f(u) is a circular function sin u or cos u. Expansion of these
will also yield expressions such as
sin sin n sin sin
ax plj yz -nt
cos cos cos cos
(there being sixteen of them if we ring all the changes on the
sin and cos). These individual expressions also satisfy the
wave equation, as can be easily seen by noting that second
differentiation with respect to x yields the same expression
multiplied by minus a 2 , and so on.
II. Some remarks on the propagation of a disturbance
through an elastic solid regarded as an ideal continuous
medium will be necessary. The proof of the statements will
be found in Love's Treatise on the Theory of Elasticity, or
similar works.
An isotropic solid has two elastic moduli ; the " bulk
modulus " or the quotient of a uniform hydrostatic pressure
by the fractional diminution in volume resulting from it ;
the " modulus of rigidity," or the quotient of a tangential
stress applied to one face of a rectangular block by the shear
of this plane past the opposite plane (supposed fixed),
resulting from the stress. All the elastic constants of the
material can be expressed in terms of these two moduli,
denoted by K and /x respectively.
If such a medium is distorted, and we represent by , 77, ,
the components of the displacement of an element of the
medium which was originally at a point #, y, z, then , 77,
ELASTIC SPECTRUM OF A CUBICAL LATTICE 191
are each functions of the three variables x, y, z* the form of
the three functions depending on the nature of the distortion.
The quantity
d J -L- ^ 4- ^
dx + Sy + dz
measures the " dilatation " or fractional change in volume
of the element which is now at the point (x, y, z). We shall
denote this by 8. This element has not only suffered (in
addition to its general displacement) a change in size, but
also a rotation, trr, whose amount is defined by the three
axial components
1 /3_ &A _ 1 /9| dj\ _ _ 1 /3, 8A
&1 - 2 \dy ~ dzj ' Wa - 2 Vai fa/' 3 ~ 2 \dx ~ dyj'
If instead of considering a static distortion, we consider a
state of vibration existing in the body, then , 17, , and of
course w^ w z , w s , d, are functions of i as well as of x, y, z.
It is known that 6 satisfies the equation
so that the dilatation is propagated with a velocity equal to
j ( K + 4 fJi/3)/p J * where p is the density of the medium.
Moreover, each component of the rotation satisfies an
equation
This involves a wave propagation with a velocity
It should be realised that the components of the displace-
ment (, 77, ) do not in general satisfy individually either of
these wave equations. They could not, of course, satisfy
both ; i.e., definite values of f , 77, , could not be propagated
with two velocities. However, there are special cases of
vibratory motion in which no rotational motion exists,
and then the displacement-components satisfy an equa-
tion similar to (19.1.2). There is one wave-velocity
{(K + 4 p,/3)/p}*. It is known that for a single plane wave
* N.B. | is in general not a function of x alone, but of all three
variables (x t y, z), etc.
192 STATISTICAL MECHANICS FOR STUDENTS
propagated under such circumstances the displacement of
an element is to and fro in a direction parallel to the direction
of propagation of the wave. In other words a pure " dilata-
tional " wave is longitudinal. There are also special cases
in which no dilatation exists, the wave is purely a rotational
one and the displacement-components then satisfy an
equation similar to (19 . 1 . 3). Under these circumstances the
displacement for a single plane wave takes the form of a
vibration in the plane of the wave. Thus the rotational
wave travels with a speed (/x/p)* and is transverse.
The energy of strain in any element of volume is, per unit
volume, known to be
...... (10.1.4)
dy dz J
19.2 The Motion of a Cubical Lattice. In the case of a
linear or superficial lattice we were able to appeal to the
results for a linear or superficial continuum, knowing that
the discrepancy between the higher frequencies of the
lattice and the corresponding frequencies of the continuum
would be rendered negligible in quantum applications by
the rapidly diminishing importance of the corresponding
average energies. We make the same kind of appeal in the
case of a cubical lattice and for the same reason ; that is the
justification for the rather long digression just finished.
Without the information contained in it, any treatment of
the lattice is apt to be very vague and not too cogent.
The lattice will have the atoms when at rest arranged in
cubical order so that there are i atoms in any row of length I
parallel to an edge. Thus in a volume Z 3 there are / atoms
where/ = i 3 . Any atom will be identified by three positive
integers r, s, u ; f being its number in a row parallel to OX,
5 in a row parallel to OY, u in a row parallel to OZ. The
suffix (rsu) will refer to this particle, and we can by methods
similar to those used for a linear lattice express each com-
ponent of the displacement of the atom (rsu) thus
ELASTIC SPECTRUM OF A CUBICAL LATTICE 193
f * * ^ r TT x s TT y u TT z
&, u = S Z Z Q wu sin -y- sin -y- sm j +
r-U-lu-l III
and similar expressions for rj rgu9 rsu where the + written at
the end is to indicate that we can have seven more series if
we like which involve terms such as cos r<{> sin sift sin u\,
cos r<f> cos sift sin ux, cos r0 cos sift cos u\, etc., obtained by
ringing the changes between sin and cos. (^, ift 9 x are written
for 77 x/l, etc., for convenience.) The Q, rsu are generalised
co-ordinates, and a little thought will show that if we used
all the eight series suggested there would be eight of them
for a particular set of values of r, s, u. As there are three
components of displacement, this would give us twenty -four
co-ordinates for each set of (r, s, u) values. In consequence,
when counting over all the terms of each series we should
have 24 i 3 or 24 / co-ordinates in all. But this is just eight
times as many as are required to co-ordinate / particles each
with three degrees of freedom. We clearly must limit the
terms in the series in some way. In dealing with the linear
and superficial lattices, we only retained sine terms, since
only with them could we satisfy the conditions at the
boundaries. The matter is not so simple here, but the clue
to a successful reduction of terms lies in suitable boundary
conditions, and in the fact that the expressions for , 77, ,
must be separable into terms expressing the dilatational
motion and terms expressing the rotational. We shall
write down the following and justify them presently.
. __ L t, L f #! cos rcf) sin sift sin u\
7=i =iu-i 1 + Q'I sin r cos s ^ cos
= X Z X . , 1(19.2.1)
y '
* . ,
+q 2 cosr(f> sm sift cos
+ 2'a cos ^cos sift sin u\
where the six quantities q l9 ...... , q'% have in general a
different set of values for each choice of the integers r, s, u.
In fact they should really be written as (qi) r8Uf etc., but to
avoid clumsiness, this complete suffixing is not used, since
no ambiguity will arise on that account. There are 6/
8.M.
194 STATISTICAL MECHANICS FOR STUDENTS
co-ordinates still involved, but assuming that q l9 q 2 > q$ refer
solely to the dilatational motion and q\, q' 2 , q' 3 to the
rotational, we arrive at three conditions which must then
hold for each set of six, and this reduces them effectively to
three independent co-ordinates. To see this, let us work out
the expressions for the dilatation and the components of
rotation.
6 = - S S Z ~~ ( rq ~*~ 5?2 "*" Uq sin r sin 5 sin
J 2 + U( l 3) cos T 9 cos 5 r cos
.. r _, , V o<j 3 u^2/ s i n r< cos 5l A cos W X
uJi - v^L^/^i / / / v / . .
1 / ( (sq 8 w? 2 ) cos r<f> sm 5^r sm u\
and two similar results.
If the 3/ co-ordinates g'j, q' 2 , q' 3 are only to supply the
rotational part of the motion, then we must have a relation
rq' l +sq' 2 +uq' 3 = Q . . (19.2.2)
for each set of values of r, 5, u. If q l9 q 2 , q 3 are only to supply
the dilatational part, then we must have the two relations
= = -. . . . (19.2.3)
r s u v
for each set of values of r, s, u. It will be seen that with
these relations, the dilatation vanishes at the boundaries
where sin <f> or sin */r or sin x is zero ; also the rotation at any
point of one of the boundary faces is around a line normal
to the face. To sum up, we can, putting q^ = rq, q 2 = sq,
q 3 = uq, write instead of (19 . 2 . 1)
If sin m }
. (19.2.4)
sm T 9 cos s v cos U X J
and two similar equations, where the relation
r ?'i + ^2 + ttff'a =
holds, so that we have reduced the co-ordinates to three
independent co-ordinates, q and any two of q' l9 q' 2 , q'$> for
each set of values for r, s, u.
The dilatation is then given by
= - 2 2 Z {(r 2 + 5 2 + u*) q sin r<f> sin s$ sin u\\
1 (19.2.5)
ELASTIC SPECTRUM OF A CUBICAL LATTICE 195
and the rotation-components by
2w l = - E Z Z {(sq' 3 - w?' 2 ) cos r<f> sin sift sin ux}
P (19.2.6)
and two similar expressions.
This is a matter of geometry and analysis. In a static
distortion, the q and q' quantities are constants ; but when
vibration takes place they are functions of t, and we appeal
to dynamics to discover the forms of the functions. A glance
at (19.1. 2) and (19.2. 5) shows that
q = <*>r*u cos K t m )
where
or oj r8U = (r 2 + s 2 + u 2 ) 1 . . (19.2.7)
c being the velocity of a longitudinal wave through the body.
Similarly the first of (19 . 1 . 3) and the first of (19.2.6)
show that
Sq'z u q*2 ^ COS (^'rsu t -~ 'rsu)
where
a>' m = ^ (r 2 + s 2 + u*)* . . (19.2.8)
l
c' being the velocity of a transverse wave. Two similar pro-
portionalities for uq\ rq'a and rq' 2 sq\ show that
q'l == a 'r*u COS (o/ r , u ^ 6 'ru)
q' %~ b' rsu cos (co' raM ^ f rsu )
2'a = c/ u cos (a)' r8U t c' nu )
where
r a' rsu + s b' rm + u c' rm = 0.
19 . 3 The Energy of the Lattice. The reader will probably
be anticipating the direction in which the argument is going,
but to complete it we must form the expression for the
energy and show that it involve^ only squares of the g, q' and
the q, q'. This may appear to be an appalling task in view
of the complexity of the expressions obtained. Fortunately
02
196 STATISTICAL MECHANICS FOR STUDENTS
it is easy to show that an enormous cancellation of terms
takes place automatically.
Let us deal with the kinetic energy first. This is obtained
by putting dots over the q and q' in (19 . 2 . 4), then squaring,
multiplying by P dx dy dz, and integrating between the
limits o and / for x, y, z. Now in such integrations all
integrands which involve sines or cosines of different multiples
of (f> (i.e., 77 x/l), or of 0, or of x> add nothing to the result,
since
r i
I . r TT x . r f TT x j . .- , f
sm - sin - dx = if r r
r TT x r' 77 x , .. j t
cos cos - dx = if r =f= r
I T 77 X T 77 X
and sin - cos -* dx = in any event.
%j
This wipes out a great body of terms, and it follows that
only the squares of individual expressions such as are written
within the brackets in (19.2.4), will yield finite amounts.
Even in these we shall find in the end no products such as
qq\, q q'& q ?' 3 , since in the case of such products we are
involved in integrations such as
sin r<f> cos rcf> sin sift cos s0 sin ux cos ux dx dy dz,
and these lead to a sero result. In the end a little inspection
will show that the kinetic energy is the sum of
,
O
over all the sets of values for r, s, u.
ELASTIC SPECTRUM OF A CUBICAL LATTICE 197
Turning now to the potential energy, we glance back at
(19.1.4), which has to be multiplied by dx dy dz and inte-
grated throughout the body. The term (< -f- 4 ju/3) 2 , in
view of the remarks just made on the ndture of the inte-
grations, clearly only involves the squares of the q co-
ordinates ; the term 4 ^(w-f + w 2 2 + m 3 2 ) involves the
squares of ' (<s#' 3 ^'2) an d such like quantities. The
quantity within the brackets { } in (19.1.4) looks as if it
might give trouble. As a matter of fact, if the reader likes
to take the trouble of working it out in detail, he will find
that the integral of it vanishes, and if he shirks the task he
can take the statement on faith. Still it looks as if the
terms referred to in the previous sentence might give
products ; for they will yield on integration the sum of
expressions such as
But the expression inside the bracket is equal to
in view of (19 . 2 . 2). Thus the statement that the energy of
the lattice involves only squared terms in the generalised
co-ordinates and velocities is justified.
So we arrive finally at a conclusion much the same as in
the previous chapter ; a lattice such as we have described
of volume Z 3 , and subject to the defined boundary conditions,
can be regarded a large complex molecule with 3/ internal
degrees of freedom which, as far as the energy expressions
are concerned, is analogous to a molecule with a number of
internal oscillators. In this analogous molecule, / of the
oscillators have each one degree of freedom, and each one
has a period belonging to the series determined by a law such
as (19 . 2 . 7). That one for instance which corresponds as
regards period to the integers r, s, u, has a direction of
motion whose direction cosines are proportional to r, s, u.
(See equation (19.2.3).) Carefully note that these remarks
are not made concerning the atoms of the lattice, but about
the oscillators of a hypothetical molecule whose energy
198 STATISTICAL MECHANICS FOR STUDENTS
function is formally identical with the formula for the energy
of the lattice. There are / other oscillators in the molecule
which have each two degrees of freedom, and each one ,a
period drawn from a series determined by ( 1 9 . 2 . 8) . The one
which corresponds to the integers r, $, u vibrates in a plane
whose normal has direction cosines proportional to r, s, u.
(See equation (19.2.2).)
19 . 4 The Statistics of a System of Cubical Lattices.
Conceive then a- system of n such cubical lattices capable of
interchanging energy one with another, via, for example,
the medium of an assemblage of gas molecules. We can
derive momenta for each q co-ordinate of the lattice by the
usual differentiation of the kinetic energy partially with
respect to each q velocity ; set up the usual machinery of a
phase-diagram of 6/ dimensions, partition the representative
points, and count the complexions in each statistical state.
Then introducing the hypothesis of quantum states, we
arrive at the conclusion that in the most probable state the
number of lattices in the system of lattices which are in a
quantum state, defined by the condition that the action-
integrals of pdq (each one integrated throughout one period
of the particular q) shall be certain integral multiples of h,
are proportional to
e -n
where e is the energy of that state and is equal to
Z Z 2 [a rm Uv nu + 2 b rltt hv' n \ . .(19.4.1)
f-1 *-l M-l [ J
the a r8H , b r8U being the 2/ quantum numbers of the particular
quantum state considered. The factor 2 in the second term
arises from the two degrees of freedom of one set of oscillators
in the analogous molecule. The average energy of a lattice
is obtained as usual by evaluating
where the 2 in the numerator and denominator indicates
a multiple summation for all values from to <# of each one
of the 2/ quantum numbers a rsu , b r8U . The symbolism has
ELASTIC SPECTRUM OF A CUBICAL LATTICE 199
become somewhat cumbersome, and it will enable the
reader to arrive at the result more easily if we temporarily
simplify the notation and consider an expression such as
S E 2 . . . (ax + by + cz + . . .)
2 2 S
a=0 6 = c=0
The letters x, y, z . . . replace for the moment the individual
hv, while a, 6, c, ... replace the quantum numbers.
Now the denominator of this expression can, as a little
thought will show, be factorised. It is equal to
(,!/"") If
The numerator can first of all be separated into a number
of expressions such as
oo oo oo
6=0 c=0
and this part of the numerator can as before be factorised
into
( S ae- a x ] ( E e~ b y \ ( E e
\a = / \6=0 / \c=0
When this part of the numerator is divided by the denomi-
nator the result is
a-0
and this is
x
g^ J
(See section 14 . 3).
Thus we see that the original expression is equal to
* -i y + z + .
200 STATISTICAL MECHANICS FOR STUDENTS
In this way it can be established that the average energy of
a lattice, viz. (19.4. 2), is equal to
S E Z\ . - hv ______ +2 ____ hl/rsu _ 1(19.4.3).
r-i f .i u =i [ exp (phv r8U ) - 1 exp (^hv' T8U ) 1 J
We can convert (19.4.3) into a definite integral just as in
previous cases. The natural frequencies of a lattice for the
q co-ordinates, which are beneath a certain value v are in
number the same as the number of positive integers the
sum of whose squares is equal to (2 lv/c) 2 . (See equation
(19.2.7).) This is the same as the number of points dotted
at the corners of unit cubes inside one octant of a sphere
whose radius is 2 Ivjc ; and so it is just one -eighth of the
volume of this sphere, i.e.,
77 /2 lv\
6 \ c I
2 lv\ 3 4 77
or -
3c
Thus the number of frequencies which lie in a range of fre-
quency v to v + Si> is
477Z 3 2 ,
- - v*8v,
c 3
and so we convert (19.4.2) into a definite integral by
collecting terms which satisfy
v + Sv > v nu > v
t i ' -^ ' --^ /
v + bv > v nu > v
to supply one element of the integral giving each of these
terms the value which corresponds to v and v and multi-
plying by the number of terms, viz. (4 TT Z 3 /c 3 ) v 2 8v and
(4 TT Z 3 /c 3 ) i/' 2 Si/. We can obviously in the final result drop
the stroke over the v and write the average energy of the
lattice of volume Z 3 or V to be
1 2
- + -
3 c
2 \ 4 77 Av 3 ,
; ) - v ---- d*> . . .(19.4. 4).
/3 / e^" 1 v ;
19 . 5 Standing Waves. The boundary conditions which
were imposed so as to give a definite character to the
problem in hand can be exhibited in another light. Reverting
for a moment to the linear lattice, it will be observed that,
ELASTIC SPECTRUM OF A CUBICAL LATTICE 201
selecting any term from the series which expressed the dis-
placement, say a r sin r TT x/l . cos a> r t, it can be written
1 . T TT , ,v , 1 T 7T , , ,v
- a r sm ( x ct) + - a r sin _ (x + ct).
If we therefore considered an unlimited string and two
progressive waves
> 1 . r TT
J
and
. ,.
r (x - ct)
r T* i i
-- a r sm (x -f
passing along it, the first in the direction of x positive, the
latter in the opposite direction, the result would be that the
appearance of progression is absent. The string would
behave as if it were divided into segments of tength Z, each
segment vibrating as, an individual in its r th upper partial.
We would have a group of " standing waves." Thus the
most general vibration can be regarded as a composition of
standing waves of different periods each wave being equi-
valent to the existence of two progressive waves of equal
amplitude travelling in opposite directions.
The superficial lattice exhibits the same features. The
expression for one harmonic of the complete vibration can
be regarded as the sum of four progressive waves such as
f = a n cos K -~ (ax + fy ct)
4 l
L
= 7 a rs COS K ? ( aX + fy + Ct )
- - a n cos *L? (ax ft/ ct)
= -a n cos --^ (ax fy + ct)
4 I
where
r
(r* +
K = (t
202 STATISTICAL MECHANICS FOR STUDENTS
The first two represent two oppositely propagated pro-
gressive waves, one in the direction (a, j8), one in direction
(a, /?) ; the second two represent waves in directions
(a, /?) and (a, j9). All four have the same speed c.
(Note that -^ a, -[- j8 are direction cosines.) Thus the lattice
with fixed edges might be regarded as a part of an infinite
lattice with four sets of widely-extended waves for each
(r, s) harmonic travelling across its surface.
Taking now a special term from the complete expression
for a cubical lattice, say
r 77 x s 77 y u TT z
cos ~- sin -~~ sin ~~
r TT x s TT y u TT z
= r a rsu cos j~ sin y- sin j cos a) rsu t
it will be found that this can be regarded as the sum of eight
progressive waves. (We drop the suffixes as unnecessary at
the moment.)
1 K7T / \
= - ra cos -y I ax + py +yz ctj
1 K7T ( \
% = - ra cos -y- (ax + fy + yz + ctj
> ! K7r f \
f = - ra cos -y- [ax py yz cti
> ! ^^^ \
f = - ra cos -y- (ax py yz + ctl
f = - ra cos y f aa; + py yz ctj
1 /C7T/ n \
= -g ra cos -y I au; -f- ^y yz + ctj
1 Krrf \
= -ra cos -y- ^ax jfy + y^ cM
= - ra cos ^ ^aa? j8y + y + cM
ELASTIC SPECTRUM OF A CUBICAL LATTICE 203
where a = (f8 + J 2 + ^ etc.
K = (r 2 + s 2 + u 2 )*.
Each of these represents a progressive wave of amplitude
ra and speed c travelling in one of the eight directions
given by ringing the changes on the signs of the direction
cosines i a, ^ /J, ^ y. In the same way the vibration-
component of 77 can be split up into progressive waves of
amplitude ^ sa, having the same harmonic factors, and
likewise for . When we associate the whole twenty -four
into eight sets of three, each trio of course involving , 77, ,
it will be found that the resultant amplitude has the same
direction as the corresponding direction cosines of the trio.
In short, the waves arc longitudinal. The transverse can be
treated in a similar manner.
Now the special feature referred to at the beginning of this
section is this : Taking a point o, y' ', z f on a face of the cube
and the point I, y' ', z' on the opposite face directly opposing
it, we see that for the longitudinal standing wave
rau = ra cos r<f> sin sift sin ^x cos (a> nu t nu )
etc.
the displacement components rj r8U9 rsw vanish while the
individual harmonic constituents of g rsu are
S TT y' U TT Z' / \
in j sin j cos I a) nu t rsu l at o, y , z
s TT y' u TT z' ( \
ra sin y sin j cos I a> rsu t * rsu J at I, y', z'.
Each of these harmonic vibrations has the same amplitude
ra sin (s TT y f jl) sin (u TT z'/l), and is in the same or opposite
phase. These remarks can be extended to the transverse
waves.
Thus we find that the behaviour of the cubical lattice
simulates that of a cubical portion of an infinitely extended
lattice through which are travelling a specially selected
group of progressive waves, one half longitudinal, one half
transverse. These waves produce standing waves in such a
manner that at points directly facing each other on two
ra sin
204 STATISTICAL MECHANICS FOR STUDENTS
parallel faces of the cube, the normal components of the
longitudinal vibration are always in the same or opposite
phases, while the components in the surface of transverse
vibration have the same character.
The drawback of this analysis of the vibrations of a lattice
(which is originally due to Jeans) for the problem in hand is
that it really concerns a continuous medium. The number
of components is really infinite, whereas for our purpose we
have to cut the series off sharp in an artificial manner since
we are thinking of a group of discrete particles. Neverthe-
less the quantum result (19 . 4 . 3) or (19 . 4 . 4) obtained from
it, was used by Debye towards the close of the first period
of quantum history to deal with the question of the specific
heat of solids and its success was so marked that when a
little later Born and Karman actually solved the difficult
problem of the real periods of a lattice proper, the improve-
ment in the agreement of calculation with observation was
not so noticeable. Born and Karman's analysis is too long
and difficult to reproduce here, but a reader with rather more
than the usual run of mathematics at his command could
consult Born's works, Dynamik der Kristallgitter , or Theorie
des Festen Zustanden. As stated, the analysis for a con-
tinuous medium was made by Jeans in 1906 in connection
with the problem of full radiation, following up an earlier
hint of the late Lord Rayleigh's. The ether being regarded,
in the familiar fashion of those days, as a continuous medium
capable of propagating transverse vibrations alone, the
energy of radiation in a volume Z 3 of it could be determined
by multiplying the average energy corresponding to a
frequency by the number of natural frequencies in a range
v to v -f- Si/, viz., 8 77 I 3 v 2 Sy/c 3 ,* where c is the velocity of
light, and summing over all the frequencies. We have seen
that statistical mechanics distributes the energy on the
average between these frequencies in the same manner as
it distributes it between the ordinary co-ordinates and
momenta of a particle system, because the motion can be
* The number of longitudinal frequencies (47rZV5v/c 3 ) is zero since c
for longitudinal vibrations in the ether is infinite, as this was considered to
be an incompressible medium.
ELASTIC SPECTRUM OF A CUBICAL LATTICE 205
analysed in terms of specially chosen co-ordinates and
momenta (in a wider sense), and to each co-ordinate there
is one corresponding frequency. This procedure has, of
course, still to be justified to the reader, but all in good time.
If then we adopt classical views, we ascribe kd to each
frequency, and find for the energy per unit volume of full
radiation in the ether (" full " in the sense that it is in
statistical equilibrium with its surroundings)
. - .<
This is the famous Rayleigh- Jeans Law. Apart from the
difficulty that the integral is infinite in value if we actually
go to the upper limit, as we strictly should do if we regard
the ether as a continuous medium, it does not fit the experi-
mental facts within any finite range of frequencies except
when well down in the infra-red. It concentrates all the
energy as it were in the upper frequencies. Even if we
artificially reduce the upper limit from oo to the highest
known frequencies (say y rays), and so evade the difficulty
just mentioned, it is clear that as between the relatively
few (though on our normal ideas of counting, enormous)
degrees of freedom say in a cubic inch of iron and the
relatively great number in a cubic inch of ether, the energy
in a state of equilibrium should nearly all be in the ether
and the iron just a trifle above absolute zero. This was
a serious difficulty for older views, even before experi-
ments on the spectral distribution of energy in full radiation
had reached such precision as to pronounce a definite
verdict against (19 . 5 . 1) in detail. These difficulties dis-
appear if we replace kd by the quantum result for the
average energy which is not the same for all frequencies,
but diminishes with frequency in a marked way on account
of the exponential term in the denominator. We then
obtain for the energy-density of full radiation
8 TT h f v 3
TT-TTm \dv . . . (19.5.2).
exp (hv/k6) 1 v '
206 STATISTICAL MECHANICS FOR STUDENTS
This is Planck's famous expression, although he obtained
it originally in a very different way (actually five or six years
before Jean's analysis). Of course (19 . 5 . 2) is a particular
case of (19.4.4) with the longitudinal velocity infinite and
the transverse velocity (here written c) put equal to the
velocity of light.
CHAPTER XX
THE SPECIFIC HEATS OF SOLID BODIES
20 . 1 Einstein's Theory for Monatomie Solids. According
to the classical statistical theory, the specific heat per gram-
atom, or atomic heat, of all monatomic solids should be
3R (5-94 calories per degree) at all temperatures. This, by
the way, is the atomic heat at constant volume while the
specific heat which is usually measured is obtained under
conditions of constant pressure and is somewhat larger than
the former, the difference being given by the formula
where a is the coefficient of thermal expansion, K the iso-
thermal compressibility, and V the atomic volume. The
theoretical s p will, of course, vary somewhat from substance
to substance by reason of differences in atomic weight and
thermal and elastic properties, but the values are round
about 6'4. Some metals keep reasonably near to this at
ordinary temperatures, e.g., silver, copper, lead, aluminium
and zinc. On the other hand, substances like boron, beryl-
lium, silicon, carbon, have values much too small at ordinary
temperatures; diamond, for example, has an atomic heat
0-75 at 50 C., and in any case all the substances show a
marked decrease of specific heat with falling temperature.
Einstein was the first to suggest that the explanation of
these facts lay in an abandonment of the equipartition
theorem. He illustrated his suggestion in a rough and ready
way by treating each atom in a monatomic solid as a har-
monic oscillator with a frequency v and three degrees of
freedom. Thus the heat energy of a solid with / atoms
would be
208 STATISTICAL MECHANICS FOR STUDENTS
and the thermal capacity would be equal to the differential
coefficient of this with respect to 0, i.e., to
where
hv
x =,*=-.
This only approaches the classical value 3R as x approaches
zero, or 6 approaches infinity ; for
On the other hand, the expression diminishes indefinitely as
x increases to an infinite value, i.e., as 6 approaches zero.
About the same time several attempts were made to evade
the difficulty along classical lines. These took the form
of assuming that with falling temperature some of the
degrees of freedom become " frozen-in," to use a picturesque
phrase, i.e., that certain linkages otherwise free, and per-
mitting some relative movement of different parts, become
completely rigid. The trouble about these suggestions lay
in the fact that if this were the case, the bodies should be
much more difficult to compress at low temperatures than
at high, and this is not so. Further, some time later, Einstein
was able to derive a value for his mean frequency v in terms
of the atomic weight, density and compressibility of the
material, and it turned out to be of the right order of magni-
tude to suit his specific heat formula. To be sure his assump-
tion of one mean frequency was bound to produce a formula
not completely in line with the facts, although a decided
improvement on Dulong and Petit's law. For one thing,
experimental work inspired by his result soon discovered
that his value for s v fell far too rapidly with temperature, on
account of the exponential term in the denominator. In
1912 Debye published a very long paper on the whole matter.
The first part consists of an investigation of the natural
frequencies of a solid continuum leading with the help of the
SPECIFIC HEATS OP SOLID BODIES 209
quantum theory to the results of the previous chapter. He
then proceeds to apply them to a lattice for which the fre-
quencies must have an upper limit. It is in this step that
the weak spot of Debye's method lies, yet his final results
constitute as great an advance beyond Einstein's formula
as the latter was beyond the law of Dulong and Petit.
We saw in section (19.4) that the number of natural fre-
quencies for a cube of volume V which lie in the range v to
v + Sv is
2\
+ pij * 2 8"-
Hence, if v m is the upper limit, the integral of this expression
from o to v m must be the whole number of frequencies, i.e.,
3/, where / is the number of atoms in volume V. Thus
4.V/1 2\
7 3Vc 3+ c'V
l V,-3JO -81
4 TT V (c 3 + 2 c 3 )
and since c and c' depend on the density and elastic constants
of the substance, v, m is calculable in terms of these quantities.
The heat energy of the body is by (19 . 4 . 4)
(c
hv* ,
dv
which is equal to
_ 9fk0 t*
J e-\
dx (20.1.2)
where we write x for hv/kd and x m for hv m /kd. The integral
is, of course, a function of x m , say </r (x m ) 9 so that the heat
energy at the temperature 6 is
W . tM.
x m
S.M. P
210 STATISTICAL MECHANICS FOR STUDENTS
The differential coefficient of this with respect to is the
thermal capacity of / atoms. This coefficient is
3 (a*) -^-j- r v~m/ ^ i 3 ^ d
_ /9/A 27/fe0
3; 3 g^ 1
.... (20.1.3).
0* I P 1 P^Wl\ 1
m J o
The expression in (20 .1.3) is a definite function of x m .
Thus Debye arrived at the conclusion, that the atomic heat
for any monatomic solid is represented by a universal
function of a pure number x m which has a characteristic
value for each solid at a given .temperature. This number
is the ratio of the quantum of energy for the maximum
frequency of the solid to the average energy (kinetic +
potential) at the temperature per degree of freedom on the
equipartition law. Obviously hv m /k has the physical
dimensions of temperature and if we call hv m /k the " charac-
teristic temperature " of a solid whose maximum frequency
is v m , then x m = 0J0, where 9 C is this characteristic tempera-
ture. Thus the atomic heat can be written according to
Debye as D (OJ0), where D (y) is a function of the variable
y defined by
If y is large, the second term in (20 .1.4) becomes
negligible, and since it is known that
it follows that for large values of y, the function D (y)
approximates to
12 7T 4 R
Hence by (20 . 1 . 4) the value at low temperatures of 8 9 is
12 77 4 R
"TV"'*
1 SPECIFIC HEATS OF SOLID BODIES 211
and so the specific heat varies as the cube of the temperature
when it approaches absolute zero. This is in excellent
agreement with the experimental facts. We would expect
the approximations at low temperatures to be good ; for in
such a condition all the energy practically resides in those
periodic motions which have the very smallest frequencies
possible to the structure. The wavelengths in the solid for
such frequencies are then so much longer than the separation
between molecules that the assumption of a continuous
medium inherent in certain parts of Debye's reasoning is
practically justified in such extreme conditions.
If on the other hand y is small enough to permit of the
approximation, e y = 1 + y, the function D (y) approxi-
mates to
y- f x 2 dx 9R,
or 3R, and so we find that the specific heat at high tempera-
tures has 3R as a limit, as it should be.
The result (20 .1.4) has been tested for about half-a-dozen
metals in a very searching manner. There is an expansion
of (e x 1 )~ 1 as a series of powers of x which is well known to
the pure mathematician, and whose coefficients have been
calculated by him as a matter of interest in other con-
nections than the needs of physical science.* This fortunate
circumstance enables D (y) to be calculated with not too
much trouble for values of y ranging from small magnitudes,
such as 0-1 to 10 or 20, or inversely to infer a value of y from
a value of D (y). From the nature of the result if the
specific heat of a solid A at a temperature 6 is the same as
that of a solid B at temperature 0' ', then 6 bears to the
characteristic temperature of A the same ratio as 6' bears to
the characteristic temperature of B. Thus from the experi-
mental values of the specific heats of a solid we should find
consistent values for its characteristic temperature. But
(20 . 1 . 1) permits us to calculate v m , and therefore C , which is
hv m /k 9 in terms of the speeds c and c', i.e., in terms of the
elastic constants of the solid. A comparison of the values
* See Chrystal's Algebra, Vol. II.. Chap. XXVIII.
P2
212 STATISTICAL MECHANICS FOR STUDENTS
of e determined by such diverse methods provides a very
clear test for the general validity of the ideas involved in
Debye's analysis. Here are some results. The values in
Column I. are derived from specific heat data, those in II.
from measurements of the elastic constants.
I. II.
Lead . . .95 ... 73
Cadmium . .168 ... 174
Silver . . . 215 ... 214
Copper . . 309 ... 332
Aluminium . . 396 ... 402
Iron . . . 453 ... 484
Since the atomic heat of a simple substance is f(0/0 c ),
where / is the same function for all the substances and 6 C is
a constant characteristic of each substance, it should be
possible to represent the variation of the atomic heat at
constant volume with temperature for all substances on the
same curve, provided the atomic heats are plotted as ordi-
nates and the values of 6/0 c as abscissae. Schrodinger has
shown that with a suitable choice of 9 C for each substance
this deduction is very thoroughly verified by experiment.
For details the reader should consult Heat and Thermo-
dynamics, by J. K. Roberts, Chap. VII.
Diamond offers a very interesting illustration of this
theory. On account of its elastic properties and its small
density the velocities c and c' in the formula (20 .1.1) are
exceptionally large. This means that the value of v m9 and
therefore of C ; is also exceptionally great. In fact, C is
1,860. Thus the temperature of diamond should attain an
abnormally high value before the value of its atomic heat
comes anywhere near to the Dulong-Petit limit. This
peculiar behaviour of diamond had been a puzzle to the
physicist for many years until the Quantum theory cleared
up the mystery.
CHAPTER XXI
THE ENTKOPY CONSTANT OF A GAS
21.1 Nernst's Heat Theorem. We are about to return to
the question raised at the end of Chapter VII., concerning
an absolute value for the entropy-constant of a gas, but
postponed for further discussion until the modifications
introduced into statistical-mechanical theory by quantum
considerations had been explained. The matter is one which
cannot be dissociated from the extremely valuable sug-
gestion concerning entropy first made by Nernst in 1906 ;
and no apology is therefore needed for a digression at this
point into purely thermodynamic theory placing before the
reader in general terms what that innovation amounted to.
In a chemical reaction or physical change which takes
place at constant volume and temperature, there is a simple
connection between the heat absorbed or evolved by the
system and the change in its free energy. Choosing 0, v to
represent the thermodynamic variables, temperature and
volume, and ctj, a 2 , ...... to represent other variables
which are involved in the reaction or change, let us denote
the internal energy, free energy and entropy by the func-
tional symbols
E (0, v, a l9 a 2 , ...... ), F (0, v, 04, a 2 , ...... ),
S (0, V,a l9 a 2 , ...... )
respectively. The amount of heat supplied to the system in
an elementary change is given by
where p is the pressure ; p is, of course, a function of
and v given by the characteristic equation of the system.
Since E = F + S, it follows that
00 ov oa r
213
214 STATISTICAL MECHANICS FOR STUDENTS
and if the reaction or change occurs at constant temperature
and volume, then
8Q = 27 !? 80, + ? -I? **
oa r oa r
r aF * or a f^\*
= L - oa r VZ I - I oa r
da r r da r \d0J r
since S = 9F/90. Thus we obtain
Hence the heat absorbed by the system during a finite
reaction or change occurring at constant volume and tem-
perature in which the a r variables vary in value from a' r
to a" r is given by the expression
Since the first term in this expression is the change in the
free energy, we have here the familiar equation of the text-
books on physical chemistry
U=A-*H . . . (21.1.1)
where U is the heat evolved during a reaction or change and
A is the loss of free energy, 3 A/90 being, of course, estimated
at constant volume.
Now it is with the free energy that the physical chemist is
vitally concerned. He can measure the quantity U, and so
the simple differential equation (21 . 1 . 1) enables him to
calculate the loss of free energy apart from an arbitrary
constant of integration. There is no clue to the value of this
constant on grounds of pure mathematics. Its value must
be obtained as the result of a suitable physical hypothesis.
Nernst suggested that trial should be made with the con-
dition that 3 A/90 diminishes in value to zero as a limit as
the temperature approaches absolute zero, i.e.,
3A
->oas0->o. . . . (21.1.2).
ENTROPY CONSTANT OF A GAS 215
The suggestion has been abundantly supported by experi-
mental results,* and so the equation (21.1. 2) is the mathe-
matical form of Nernst's heat theorem, and by it is deter-
mined the " chemical constant " of any reaction and the
ambiguity in the solution of (21 . 1 . 1) removed.
But from what just precedes, we can clearly write the
condition (21 . 1 . 2) in the form
9 F (0, V, a' v a' 2 ...... ) 8 F (fl, v, a* l9 a%, ...... )
if = o ; or
S (M,a'i,a' 2 , ...... ) - S(0 9 V,a* l9 a" 2 , ...... ) (21 . 1 . 3)
if r= o. In words, the difference between the entropy of
the system before and after a reaction gradually disappears
as the temperature at which the reaction occurs is reduced to
the absolute zero as a limit.
21.2 The Entropy-Constant and the Vapour-Pressure-
Constant. As we saw in Chapter VII. the entropy of a
gram-molecule of a perfect gas containing one type of mole-
cule is
s p log9 -Rlogp + K . . . (21.2.1)
where * is a constant of integration, which is of no import-
ance in dealing with the change of entropy between two
states of the gas, but assumes quite a new significance when
Nernst's theorem is considered in connection with it. Thus
let us assume that at high enough temperature the vapour
of a liquid is in such a condition that (21 . 2 . 1) is a good
enough approximation to its entropy at temperature 9. At
the same temperature let </> (0) be the entropy of the liquid ;
then
where s is the specific heat of the liquid. Of course in <j> (0)
must also occur an undetermined constant of integration ;
* See System of Physical Chemistry, Vol. II., Chapter XIII. ; W. C.
McC. Lewis.
216 STATISTICAL MECHANICS FOR STUDENTS
but this constant is clearly settled by the value we choose
for K in (21 . 2 . 1) ; for
#(0)+==a,log0--Rlogi> + jc . . (21.2,2)
since the two entropies must differ by L/0, L being the heat
of vaporisation.
Let us turn for a moment to the well-known thermo-
dynamic relations which hold for the change of state from
liquid to vapour, and which are associated with the name of
Clapeyron. They are
d L
dL
In the second, s l is the specific heat of the saturated vapour.
Now in the condition of saturation at temperature 6 + 80,
let the pressure be p + Sp, and the specific volume be
v l + 8v v Conceive the vapour to expand to a specific
volume v l + ^i + S'^u so that its pressure is restored to
the original value p, the temperature still being 6 + 0, so
that the vapour is unsaturated. If the temperature be so
high that the conditions approximate to those of an ideal
gas, then by Boyle's law
(P + 8p) K + 8*1) = P(Vi + 8*1 + 8'*i),
or practically
v 8p p S'v l
Now from the definition of s l it follows that
8't>
^=i+P^-
Therefore
Since v 2 is small compared to v l9 this can be practically
written
ENTROPY CONSTANT OF A GAS 217
dp
S P =S! + ( Vl - *,) ^
= i+ j by (21. 2. 3).
Hence the second of the relations above becomes
dL
where s, the ordinary specific of the liquid can be regarded
as practically equal to s 2 . Thus (21.2.5) can be regarded as
valid at sufficiently high temperatures. But the first relation
(21.2.3) can under these circumstances be written
dp pL
d log p L
-
To connect these two approximate results (21 .2.5) and
(21.2.6) with our previous considerations of entropy, we
must integrate them. Thus (21.2.5) yields
f
L = L + s p O \s (x) dx,
Jo
where L is a constant of integration which may be called
the " latent heat at absolute zero " without necessarily
implying any physical reality for a change of state actually
at this temperature. Of course s p is a constant approxi-
mately equal to 5R/2, but s is a function of temperature.
The integral on the right-hand side is quite definite, and is
equal to the increase in energy-content of the liquid *
between absolute zero and 0, this increase being the actual
energy-content if we consider the energy-content to be nil
at absolute zero. But in any case it is a definite function
ctf 0, say E (9), and it is implied that E (o) = o. Thus
L = L +s/~E(0) . . . (21.2.7)
* If the condensed system at zero is really a solid, no ambiguity arises.
The energy-content then contains a constant term (the latent heat of
fusion) in addition to the integral.
218 STATISTICAL MECHANICS FOR STUDENTS
Turning to (21 . 2 . 6) we find
dlogp
(21.2.8)
where y is a second constant of integration ; for the moment
we pass over the point as to the possibility of an indefinite
quantity arising at the lower limit of the integral since
E (0)/0 2 is apparently an indeterminate quantity when
6 = 0. It follows that
s p log 9 - R logp = ^ + ^ dx - Ry
U i X
Hence by (21.2.2)
which by (21.2.7)
We can now derive the value of the entropy of the con-
densed substance at the absolute zero of temperature. It is
Lt E (9)
e=o-~r +K ~ s p-^
By definition the first term on the right-hand side is s
where s is the specific heat of the condensed system at
absolute zero * ; thus
< (o) = K - 8 p - Ry -f 8 .
We are now in a position to see the bearing of Nernst'tf
theorem on this analysis. Cases arise in which several
allo tropic forms of the condensed solid or liquid exist. Each
form will have its own individual values of energy-content,
latent heat, etc., at a given temperature 6. Hence since p,
* Of course, although E(0)/0approaches a definite value as 6 approaches
zero, it does not follow that E(0)/0 2 does so ; hence the possibility of
ambiguity in the integral f ~&(x)jx z .dx.
ENTROPY CONSTANT OF A GAS 219
the pressure of the vapour, depends only on 6, it might
readily be inferred from (21.2.8) that the vapour-pressure
constant y would depend on the particular allotropic form.
But any chemical reaction in which one form is converted
into another would, according to Nernst's theorem, change
the entropy by an amount which tends to zero as the
temperature at which the reaction is carried out is reduced
in value. This means that <f> (o) is the same for all allotropic
forms. Now experiments on the part of Nernst and his
collaborators were gradually convincing them that in point
of fact the specific heats of simple solids and liquids tend to
zero as 6 approaches zero. Accepting this as an experi-
mental fact, it appeared that the common value for the
entropy at zero of any of the various condensed forms is
K - s p - Ry . . . .(21.2. 10).
Experiment was moreover indicating more than the simple
fact that s or d E (8)/d9 tends to zero as approaches
zero ; it was suggesting that the approach towards the limit
is so rapid that E (9)/8 2 also tends towards zero, thus making
the integral in (21 . 2 . 8) a perfectly definite quantity without
any indeterminateness arising at the lower limit. This is,
of course, quite in keeping with the theoretical discussions
of Einstein, Debye, etc., which, however, were historically
later. That being so, y has a definite value for each form of
the condensed material. But by (21.2. 10) K Ry must be
independent of the particular form, and since AC, whatever
value it receives, is only dependent on the vapour, the con-
stant y is therefore the same for all the allotropic forms of
the condensed material.
It was at this point that Planck stepped into the dis-
cussion. Thermodynamics, even with Nernst's theorem,
was still unable to assign any value to (21 . 2. 10) , the value
of the entropy at zero. The constant y can, no doubt, be
experimentally determined by applying (21 . 2 . 8) to vapour-
pressure measurements, but, of course, the entropy -con-
stant of the vapour, K, is, on thermodynamic grounds alone,
entirely undetermined. By applying the quantum hypo-
thesis to the statistical-mechanical considerations of Chapter
220 STATISTICAL MECHANICS FOR STUDENTS
VII., Planck pointed out that there was considerable support
for the view that the value of the entropy at the zero of
temperature tends to zero as a limit. If this were so, this
meant that one must assign a definite value to the entropy-
constant, K, of a vapour or gas, viz., Ry + s p9 where y is the
experimental vapour-pressure constant, and s p the ideal
specific heat at constant pressure. It remains to show how
Planck was guided to this conclusion. His arguments were
not regarded as quite convincing at first, and an extremely
interesting discussion on the statistical-mechanical side of
this matter went on for several years. We shall endeavour
to summarise these investigations in the following chapter.
NOTE. For the sake of the readers who are familiar with
the System of Physical Chemistry, by W. C. McC. Lewis, it
may be as well to point out that the y of the text above
is connected with the " characteristic constant " of the
vapour, denoted in that work by the letter i, by the simple
relation
y = i + log R.
For since
p = R C 9,
where C is the concentration, it follows that
which is the expression on page 76 of Vol. II. (2nd edition).
The logarithms are, of course, to the Napierian base, e.
The usual numerical values of the Nernst constant are
determined on the understanding that 10 is the logarithmic
base. In that case we must write the value for the logarithm
of the vapour-pressure as follows
i L o , s p i a 1 f * E (x) ,
-^
, y i + log R
where c = - = ' - .
2-3023 2-3023
CHAPTER XXII
THE ENTROPY CONSTANT OF A MONATOMEC GAS AND
STATISTICAL MECHANICS
22.1 The Magnitude of the Phase-Cell. The reader will
recall certain remarks made at the beginning of Chapter
XVI., in which was mentioned the question of adjusting
probability calculations for a system whose particles may
pass from a condition involving quantum paths to one in
which the paths are not so restricted. No answer to such a
problem can be evolved from purely mathematical a priori
considerations ; it is necessary to introduce a definite
postulate and test its results by experiment. It was stated
that one plausible hypothesis assigns even for a non-
quantised state a fundamental size to the phase-cell depending
on a suitable power of h ; in the case of particles with the
usual three degrees of freedom for translatory motion it will
be the third power. Now such a postulate naturally leads
one to speculate if a similar specification of the size of the
phase -cell may not prove convenient even when alternations
between quantum and non-quantum conditions are not in
question. In point of fact, very early in quantum history
Planck introduced such considerations into the statistical
treatment of a moiiatomic gas. We hinted as much in
Chapter VII., where a symbol g was used pro tern, as indi-
cating some definite magnitude having the dimensions of
action cubed. There is certainly none of the obvious
support for this view, which is supplied to the analogous
hypothesis for internal vibratory motions where spectroscopy
has provided such powerful aid. Nevertheless, discussion of
this suggestion of Planck's proceeded apace, and, beginning
with two rather famous papers by 0. Sackur and H. Tetrode,
a considerable volume of literature poured out dealing
with this matter, gradually converging to the view that g
221
222 STATISTICAL MECHANICS FOR STUDENTS
not only exists as a definite magnitude, but that its value
is A 3 . Indeed, in some quarters attempts have been made to
justify the postulate that even for the translatory motion of
molecules in a gas system, quantisation of paths exists, and
that experimental facts to support such a claim might be
found at sufficiently high concentrations or at very low
temperatures. In the former case the space available for
molecular motion becomes restricted and molecular motion
would have a zig-zag character which might be regarded as
a vibration whose central point is gradually shifting. In the
latter case the average molecular energy becomes very small.
No very definite experimental results to prove such ' ' degene-
ration of gases " are available, but it must be admitted that
the conditions under which quantisation would become
apparent would be very extreme, and it cannot be said that
the claim is absolutely disproved as yet. The following
rather artificial analysis shows how one might formally
express this view. Regard the gas as enclosed in a cubical
box of side 1. Disregard intermolecular collisions and con-
sider all collisions as between molecules and the sides. This
is a kind of vibration on the part of each molecule with three
degrees of freedom the amplitude for each degree being L
The action-integral for the component motion parallel to
one edge (say the axis of x) is
\mv x
J n
dx,
which is equal to 2 m v x /, and similarly for the other com-
ponents. Thus, corresponding to three integers r 1? r 2 , r 3 ,
we would have a particular velocity of translation for a
molecule given by the conditions
2 m v x I r 1 h
2 m v y I r 2 h
2 m v z I = r 3 hy
so that the energy of a molecule would have one of a series
of discrete values such as
w . 2 , 2 , 2)
8 m l z ( x 2 3 ;
CONSTANT OF A MONATOMIC GAS 223
each value corresponding to a special choice of the integers
r i> T 2> r 3
Quite apart from these rather speculative considerations,
indirect experimental evidence is available for choosing a
definite size for a phase-cell even in the case of gases and
putting it equal to A 3 . It exists in connection with the pro-
blem discussed in the last chapter.
Reverting to Chapter VII., it was shown that if the number
of representative points in the respective phase-cells in the
most probable state are denoted by v l9 v 2 , v 3 , ...... v c , then
, f 5 , fl , , , (2 TT m k)*k , 3)
- S v r log v r - n \- log e - log p + log -i - i + _L
r = l I* 9 A)
(See equations (7.1.3) and (7.2.1)).
If W m stands for the complexion-number of the most
probable state, this means that if we adopt the suggestion
forgr
(5
* (log W m n log n) = R - log 9 - log p
. . . (22.1.1).
The reader will recall that in discussing the statistical
basis of the second law of thermodynamics in Chapters VI.
and XL. we selected the expression on the left-hand side of
(22 . 1 . 1) as the entropy of a system. Nevertheless from the
classical point of view it is clear that k log W m any con-
stant would serve just as well. However, a reference to the
conclusion of the last chapter will also remind him of Planck's
suggestion that the entropy of a condensed system should
be so chosen as to vanish at absolute zero. One necessary
conclusion of this would be that the absolute entropy of any
substance in this sense would be simply proportional to its
amount. Thus the entropy of a quantity of gas would be
proportional to n, and this consideration seems at once to
justify the removal of k n log n from k log W m , as the
expression on the right-hand side of (22 . 1 . 1) is certainly pro-
portional to n. But a little reflection raises doubt once more
since the further addition or subtraction of a multiple of n
224 STATISTICAL MECHANICS FOR STUDENTS
(which is a constant) would still leave the expression for the
entropy proportional to n. On reflecting still more, we are
led to consider how the removal of k n log n arises in the
analysis. It takes place when we write for W m the expression
n n
and then divide W m by n n , so as to clear n out of the
expression for " probability " and to obtain a logarithm of
probability which is proportional to n. But the expression
just written for W m is an approximation ; why not clear n
out of the exact expression for W m , viz.,
n\
If we do so, we would choose for the entropy not k log (WJn n )
but k log (WJn !), i.e.,
k (log W m n log n + n).
Now this line of reasoning is manifestly weak ; neverthe-
less it leads to a result which can be tested by experiment.
The entropy now becomes
^e-lag P +1 0g VW k + *l . (22.1.2).
/I 2^ I
If this be so, then we have at once ascertained a value for,
the entropy constant of the last chapter, viz.,
-P , (2 77 m k)l k , 5R
* = Rlog<~ A , - + a
and from the conclusion there established that the vanishing
of the entropy of the condensed system at absolute zero
makes K equal to Ry + s p9 we arrive at the equality
r = log (2,.U _ _ (22 , 3)
since s p = 5R/2.
But y is a vapour-constant which can be determined by
experiment. In (21.2.8)
CONSTANT OF A MONATOMIC GAS 225
is an integral which can be calculated from specific heat
measurements by the method of quadratures, or by the
expansion of E (6) as series of powers of 9 whose coefficients
can be obtained from the experimental data. Thus (21.2.8)
can be used in conjunction with specific heat and vapour-
pressure measurements to find y. Equation (22 .1.3) can
be written
, (2 77 M) 3 R*
y = tog -^-'
where M is the molecular weight of the gas, n the number of
molecules in a gram-molecule of gas (6-06 x 10 23 ) and R is
the gram-molecular gas constant (8-32 x 10 7 ). Thus
y=y + l-5logM . . . . (22.1.4)
where
3 5
y = ' log 2 77 + log R 4 log Ti 3 log h
2i 2i
= 10-17.
If we adopt the base 10 for the logarithms as is general in
the actual calculation of Ncrnst's constants, we find that
10-17 , , r , , T
c = -f 1-5 Iog 10 M
2-3026 tol
= 4-41 + 1-5 Iog 10 M . . . (22 . 1 . 5).
The experimental determination of c in the cases of
hydrogen, mercury and argon give values approximating to
4-3, 4-4, 4-45, respectively for the constant term in (22 .1.5).
The reader will recall that at sufficiently low temperatures,
hydrogen behaves as a monatomic gas.
Despite this a posteriori justification by experiment no one
can feel that the initial argument for k (log W m n log
n + n) as the entropy in preference to many other expres-
sions apparently as plausible is very satisfactory. It is
round this point, rather than the choice of h 3 as the magni-
tude of the phase-cell, that the discussion has been most keen.
Planck himself has been most persistent in endeavouring to
justify the division of W m by n \ by considerations involving
no other state than that of gas. His latest views are given
at some length in the fifth edition of his famous book Theorie
226 STATISTICAL MECHANICS FOR STUDENTS
der Wdrmestrahlung. Briefly they amount to a statement
that if we " quantise " the movements of the molecules, it
does not matter to which molecule we attach a particular
set of values for the quantum numbers r l9 r 2 , r 3 , and as there
are n \ permutations of n points possible, division by n !
simply embodies this indifference as to the individuality of
the molecule. Nevertheless many investigators regard such
arguments as fundamentally weak and hold that only by
considering some process in which the number of molecules
in the gas can be varied, can we settle without ambiguity
the dependence of the entropy expression on n. Such a
process can be found in the sublimation of a gas from a
solid state ; we can regard the molecules in the solid as in
quantum states without postulating quantum states for the
gas. The following is a short sketch of a line of argument of
this type due to 0. Stern.
22 . 2 Stern's Treatment of the Entropy-Constant Problem.
Stern's paper is lengthy and rather involved. It falls
naturally into two parts. In one of these, the vapour
equation (21 . 2 . 8) derived in the last chapter is adapted to
suit the view that the internal energy E (6) of the condensed
part of the system is determined by quantum considerations ;
i.e., thermodynainical reasoning goes into co-operation with
the hypothesis that quantum states exist in the condensed
substance (which is regarded as a cubical lattice such as we
dealt with in Chapter XJ X.), but no hypothesis that quantum
states exist in the vapour is introduced. The result of this
train of reasoning is the equation
,
r=* 1 2 fj //jiA
logp = --- _--_ + L log + Slog (jgj + y(22.2. 1)
where v v ^ 2 , ...... , v^ are the 3/ natural periods of a
lattice of / atoms as given by (19 .2.7) and (19 .2.8) and v
is their geometric mean in other words
" 3/ - Vi . v 2 ...... v 3/ . . . . (22 . 2 . 2)
R is written for fk.
The other part of the paper is the derivation by statistical-
CONSTANT OF A MONATOMIC GAS 227
mechanical methods of a vapour-pressure formula for a
cubical lattice and its vapour in statistical equilibrium. In
this part quantum considerations are not involved, since the
temperature is regarded as high enough to permit of the use
of classical statistics. In this part Stern finds that
. (22 . 2 . 3)
In this, W =fw, where w is the work required to bring one
atom from complete rest in the solid into the vapour. No
constant of integration appears in this equation ; the mole-
cular model is definite, and so gives the absolute value of the
vapour pressure. Thermodynamics, on the other hand, only
gives dp/d9 or d log pjd9 ; it is for this reason that a constant
of integration appears in (21.2.8) and (22 .2.1). A com-
parison of (22 . 2 . 1) and (22 . 2 . 3) now yields
W^L + l 7 ^r
r-l *
and the result which we are anxious to justify, viz.
, (2 77 ra k)% k
^ ] g A*
We shall briefly outline the development of each part of
the paper.
I. The Thermodynamical Part. In (21 . 2 . 8) let us write
for a lattice of / atoms
where, as usual, //, =
Our task is now to integrate
% ^ : dx
r =ix 2 {exp (hv r /kx) 1}
from o to 0. Considering one term, we have
dx
exp (hv r /kx) 1
dy
228 STATISTICAL MECHANICS FOR STUDENTS
where y = hv r /kx and 17 = hv r fk6. This is equal to
f f e y
*l \-T \~
J, [e y 1
oo
At the upper limit, y = GO , log (e y 1), is practically
log e y or y, and so the integral is equal to
kr) k log (& 1 )
7.
Now
log (e^ - 1) - log ^ hv + f /^ 2 ^ ^ 2
log /it h v + log / 1 + - p. h v + ...... j .
If the temperature be sufficiently high, p, h v is small
enough to allow this to be written
log (e hv 1) log p, h v + - p h v.
2i
Hence it follows that
and thus equation (21.2.8) becomes
T f 7 If 7 27 ^i-
L 5 ^ v Av r r-1 r
log p == H log -) 27 log -
JL\iC7 2i JL\ j.,-,! A/v
^ ilv
L + 27 -r-
using (22 .2.2) and remembering that R = /fc. This estab-
lishes equation (22 . 2 . 1).
CONSTANT OF A MONATOMIC GAS 229
//. The Statistical-Mechanical Part. The second part of
Stern's paper will probably cause more trouble to follow,
since Stern uses the statistical method associated with the
name of Gibbs. This method will be explained in Chapter
xxiv., but even at this stage it should not be beyond the
power of the reader to grasp its essential idea. Indeed we
have already come very near to its use in Chapters xvin.
and xix. Recall the fact that we regarded a lattice-system
of / atoms as a huge molecule and considered the statistics
of a large number, n, of such systems in an enormous gaseous
envelope at a given temperature a " temperature-bath "
in fact. Thus we were dealing essentially with a " Gibbs
ensemble " (or assembly) of n molecular systems. It is this
idea that we shall exploit still further here.
The system is made up of A atoms in an enclosure. Of
these a atoms will be in a gaseous state, and / will be in a
solid lattice formation, so that A = a + /, but, of course, a
and / can vary individually. This system is to be regarded
as the " huge molecule." An ensemble of these systems,
n in number, is supposed to be in the temperature-bath at
temperature d. Thus the systems will exchange energy and
each system will have a history or travel along a " path "
in a phase-diagram which will involve the representation of
3A co-ordinates and 3 A momenta. The system co-ordinates
will be x v y v z l9 , x a , y a , z a , q l9 &, ? 3 , q sf
where X T , y r , z r , are the Cartesian co-ordinates of the r th
molecule in the gas, and q^q^ , # 3 / are 3/ " normal "
co-ordinates of the lattice, as explained in Chapter xix.
The momenta of the system are 1? ??, x , , f a , 7? a , a ,
jPi, Pz, p*f where & = mx n rj r = my r , r = mz , and
p r is the differential coefficient with respect to q r of the
kinetic energy of the lattice which was shown to be a quad-
ratic function of the q r involving only squared terms. A
phase of a system is defined by the assignment of definite
values to x v # 3/ , g l9 , p 3f . An extension-in-
phase of the system is defined by a statement that the phase
of the system lies between x l9 , q 3f , g v
and x^ + 8x, q 3f + Sq 3f , ( + 8^, p 3f
Now the probability that a particular system out of the
230 STATISTICAL MECHANICS FOR STUDENTS
great number n of these systems is in an elementary phase-
extension, so defined, is
I exp (- p, E) So?! ...... 8z a , 8g x ...... S# 3/
Sfi ...... S>! ...... 8^ 3/ . (22.2.4)
where I is some constant, and K is the energy of the system
in the central phase of this extension which is practically
...... + , ( }
2m ' '
where (/>(#, >) stands for the quadratic function which is
equal to the sum of the kinetic and potential energies of the
lattice, and w is the work required to bring a molecule at
rest from the lattice into the gas.
Having settled these preliminaries, let us work out the
probability that the system is in such a condition that a
particular atoms are in the gas, and the remaining / par-
ticular atoms are in the lattice. It will be the sum of the
probabilities (22 . 2 . 4) for all possible phases, i.e.,
if ...... I" ocp (- /x B) dx l ...... dpv . . (22 . 2 . 5)
over all phases.
Our immediate object must be to perform this integration.
In the first place (22 .2.5) splits into the product of two
multiple integrals
J
dxi ...... dz a d& ..... d a . (22.2.6)
and
j ...... J exp [ ~ fji <f>(q, p)} dq l ...... dp 3f . (22.2.1)
Since [*exp \ - p 2 /2m ! d = ( ) [ V*" dx
j jj \ p, / j ~
it follows that the expression (22 . 2 . 6) is equal to
e)^ . . . (22.2.8)
CONSTANT OF A MONATOMIC GAS 231
where V is the volume of the gas, which can practically be
considered the volume of the enclosure.
To work out (22 .2.7) does not require so much " grind "
as one might imagine, if one refers back to certain expres-
sions in Chapter xrx. If the reader will look at (19.1. 4),
(19.2.5), (19.2.6), (19.2.7), (19.2.8), and bear in mind the
substance of section (19 . 3), he can, if he takes a little trouble,
convince himself that the energy (f>(q, p) of the lattice in a
particular phase can be expressed in the form
2 '
where co l5 , o> 3 y are the 3/ natural pulsances of the
lattice and we have supposed that the co-ordinates q-f
q 3 j have been changed by a constant multiplier to such
values that the coefficient of p^ in the kinetic energy is half
of unity. This makes the calculation of (22 . 2 . 7) fairly
simple, for since
f x /I \ /2\ f
exp I ~LL p 2 } dp { - ) e~ x * dx (2 TT k 6)*
L,/ V 2* 1 ) \p/ J-oo V '
s* rj / i \ / o /. /} \ '
i / - 1 oo\7 ( ^ 77 A, 17 )
and exp ( ~ ^ ^ q ) dq *
J _ ^ \ 2 / 60
it follows that the expression (22 .2.7) is equal to
f~(277/<;6
^ rr r i/ x . v 2 i/ 3/
(/" f)\ 3f
-f) (22.2.9)
v being the geometric mean of the 3/ frequencies v l9 v 2 ,
Combining (22 . 2 . 8) and (22 . 2 . 9), it would seem that the
probability that the A molecules in the enclosure are so
* The integration is not actually from q = <x> to q = -+- <x> ; but we
have on several occasions used the same procedure, since the contribution
to the integral of the range of q beyond the actual extremes is negligible.
232 STATISTICAL MECHANICS FOR STUDENTS
situated that a particular molecules are in the vapour and
/ particular in the solid is
But in this conclusion we have overlooked one point.
We have implicitly assumed that there are not only / par-
ticular molecules in the lattice, but that they are disposed
in a particular way, each molecule oscillating about a par-
ticular position of equilibrium. With these / molecules we
can make / ! different arrangements out of any one which
we have just considered, by permuting the molecules
precisely as if they were / balls in / " pigeon-holes." It
follows that the complexions are / ! more numerous than we
thought, and the probability for a particular molecules in
the gas and / particular in the solid is
?? /If fl\ z f
~ I J . .(22.2.10).
The last step in the probability calculation involves the
removal of the epithet, " particular. " There are A I/a !/!
ways of choosing a molecules to go in the gas and / in the
solid ; so the probability that there are a molecules (any
molecules) in the gas and / in the solid is
At * a /1fti\
Ll e -i* V a (2irmkO)~*( )*f . (22.2.11).
a ! \ v /
This is a function of a and / (really of a only, since a + /
is constant) and our business is now to find when it is a
maximum, as this will give us the most probable distribution
of the molecules between gas and solid, i.e., the distribution
which we will meet in experiment. To find this, we put the
differential coefficient of the expression (22 . 2 . 11) with
respect to a equal to zero or rather more easily, the
differential coefficient of the logarithm of it. Thus we find
t o
-- log a ! p, w + log V + log (2 TT m k 0)
da 2
3 log k + 3 log v =
CONSTANT OF A MONATOMIC GAS 233
(remember that df/da = 1). Putting log a \ equal to
a log a a, we find
i , i TT w , 3 ! /27T m\ , , n
-loga+logV- +-log(_) +31ogv = 0.
But the pressure in the gas is given by
so that
log p = log a log V + log (Ic 9)
= - p + | log (27T m) - I log ifl + 3 log v,
which is just (22 . 2 . 3).
CHAPTER XXIII
ENSEMBLES OF SYSTEMS. I
23 . 1 The Probability Postulate and Dynamics. Through-
out all the reasoning so far has run the implicit assumption
that the probability of the occurrence of a particular state
of a system is proportional to the number of complexions
embraced in that system. It now behoves us to turn our
attention to any justification which can be found for this
postulate. As we pointed out in an earlier passage, the
complexions of a system can hardly be said to follow one
another in the same manner as the complexions of a group
of coins or dice follow one another in the chance of the cast.
In short, we have to see what dynamical principles have to
say in this matter. When it comes down to " brass tacks,"
the reader may as well realise at once that we are on very
doubtful ground indeed if we insist on rigorous logic unless
we abandon the attempt to be definite about one system
only and enlarge our field of view to embrace the lives of
many systems each having the same dynamical character,
but each having its own individual history which is distinct
from those of its fellows.
Paradoxically enough, if we attempt a similar enlargement
in the case of coin-tossing or dice-throwing, we are entirely
in the dark experimentally. Many people have tossed
pennies and found that out of a long succession of attempts,
practically half are heads and half tails ; but the author is
not aware of any occurrence in which several thousand
people threw pennies in the air simultaneously and observed
the distribution of heads and tails at each throw. Indeed,
we should have to imagine such a crowd of people each
tossing several dozen pennies and observing how far the
proportionality of probability to complexion-number is
obeyed when the complexion of each system of pennies is
234
ENSEMBLES OF SYSTEMS. I 235
recorded and the facts for the " ensemble " or assembly of
the systems compiled. Yet if instead of dealing with one
dynamical system we consider an ensemble of systems, we
are on very firm ground experimentally and we proceed to
show why.
We shall begin with a very simple system indeed our old
friend, the uniformly accelerated falling particle ; it is a
system with one degree of freedom ; i.e., its position is given
by one co-ordinate, viz. , the distance below some definite level.
But we want to consider an ensemble of these systems ; so
we shall visualise a shower of rain with the resisting air
conveniently removed and arrange that each drop has its
own line of fall with 110 drop exactly in a vertical line with
another. Under these circumstances many drops may at
one moment be at the same level, i.e., have the same value
of the co-ordinate, which we shall denote by q, but we must
conceive that they have not all the same value of velocity
or q (q is written for dq/dt) ; in other words, these drops have
fallen from different heights at such instants as to place
them now (but not before or after) at the same level. Look
at this another way ; we know from dynamical theory that
q=a + bt+~gt 2 . . . . (23.1.1)
In this equation a and 6 are constants of integration ; their
values are arbitrary ; they vary from particle to particle
and give an individuality to each member of the ensemble.
But g is common to all members. It is determined by the
field of gravity i.e., the external bodies acting on each
system affects them all alike. We call it a " parameter " of
the system. It is obviously possible to choose many sets of
values of a and b which give the same value of a + bt for a
given value of t ; the particles involved will all have the
same value of q at the time t. But since
q =b+gt . . . . (23 . 1 . 2)
and since 6 is not the same for this group of particles, their
velocities are different at this level. In this way we arrange
that no two particles have the same phase at any given
moment, phase depending on both position and velocity.
236 STATISTICAL MECHANICS FOR STUDENTS
Suppose then we represent this state of affairs on a phase-
diagram. We shall draw as usual two rectangular axes,
OX, OY, and represent the phase of each system at a definite
instant by a point P, the x co-ordinate being equal to q, the
y co-ordinate being equal to the momentum, mq. As the
system moves, i.e., as the drop falls, the phase-point P
moves along a stretch of a parabola in the phase-diagram ;
this is the " path " or " trajectory " of the representative
point. This point has a velocity in the phase-diagram,
whose components parallel to OX and OY are q and p
respectively. For the moment denote these by u and v, so
that u q, v p.
Now it is not difficult to express u and v as functions of x and
y. Thus the energy of any system of the ensemble is equal to
mq* + mg (q q),
where q is a constant (the same for all systems of the
ensemble). This can be written
^ i ^P^ + mg(q -q) . . (23.1.3).
Represent this function by E(g, p). It is easily seen from
(23 . 1 . 2) that
3<Z '
or u ="-^yl .... (23.1.4)
V =
Bx
D , . Su d 2 E(x,y)
But since = i-^L'
ox ox oy
and
dy dy dx
it follows that
ENSEMBLES OF SYSTEMS. I 237
This simple relation embodies a result of supreme import-
ance in our statistical considerations. Since we are con-
sidering an ensemble of systems, the phase-diagram will be
covered with a crowd of points or dots, each one moving
along one member of a family of parabolas. We can conceive
the drops of rain to be so numerous and so many phases to
be represented that the representative points are at any
moment closely packed together in the phase-diagram,
forming what is practically a dense " cloud " of points
similar to a two-dimensional continuous fluid, and, of course,
as the drops fall the cloud moves in the phase-diagram.
Imagine that any moment the cloud fills a portion of the
phase-diagram within some closed geometrical curve. Later
its position has changed ; it occupies a different region
within another curve, most likely of different shape, but the
second region has the same area as the first ; that is the vital
point. As each system of the ensemble passes through its
successive positions and momenta, the cloud of representative
points moves about on the phase-diagram, always occupy-
ing the same extension-in-phase. To prove this, all we require
is equation (23 .1.5). This, and, of course, a corresponding
wider theorem, for systems with many degrees of freedom,
is the great contribution of theoretical dynamics to statis-
tical mechanics. Its implications we shall elicit presently.
The proof for the simple one -degree system is not par-
ticularly troublesome.
Thus conceive an elementary rectangle of area 8x Sy in
the phase-diagram whose centre is at the point (#, y), and
let us calculate the net number of representative points of
the cloud which move into the rectangle in time St, which
elapses after a certain instant denoted by t. At that
instant there will be a definite distribution of density over
the phase-diagram for the R-points.* We shall denote it
by p, where, however, we must observe that this density is
a function of x, y and t, and should really be symbolised by
* We must distinguish between points in the phase -diagram in the
ordinary sense and representative points which are conceived to have a
kind of substantiality and to move about. To signalise this we shall call
the latter " R-points."
238 STATISTICAL MECHANICS FOR STUDENTS
a functional form, such as/(x, y, t) ; for we are not assuming
that the R-points are uniformly distributed at any definite
instant, not even over the area in which we considered them
to be originally placed ; nor do we assume that the density
remains unchanged at a definite point of the phase-diagram
throughout all time. It should also be remembered that
u and v are functions of x and y also, and should be written
t(x, y), MX, y)* (See (23 . 1 . 4).)
Consider that side Sy of the elementary rectangle which is
nearest to the axis OY. Taking the value of u at its mid-
point (x o Sx, y), we see that the number of R-points
which cross this side into the rectangle in time St will at the
instant t lie in a parallelogram whose side is 8y, and height
St . <f> (x 7! Sx, y) ; for, of course, the component v of the
velocity contributes nothing towards transport across a line
parallel to OY. We can conceive St to be so small that this
height is small compared to Sy, and so the density of the
points in it at the instant t can be taken to be/ (x I Sx, y, t).
Thus the number of R-points entering the rectangle by this
Sy side in time St, is
Sy /(* - 2 > y> ^ i (* - i ^ y) - ( 23 l 6 )
Similarly the number which leave the rectangle across the
opposite Sy side in time St is
Syf(x+^8x,y,t)8t<f>(x + ^8x,y) . (23.1.7)
In precisely the same manner the number which enter
and leave the rectangle by the lower and upper 8x sides
respectively, are
8xf(x, y + J Sy, t) St t(x,y + \ Sy). ' -
The mathematician is prone to lifting his eyebrows at this
point, and, when it comes to mathematical rigor, he is justi-
fied in expressing doubt about our procedure ; but life is
short, and, in this book at all events, we cannot enter into
all the mathematical niceties of proof. The reader can rest
assured that the procedure can be justified if certain restric-
* In general dynamical reasoning u and v may also involve t ; but we
are excluding cases where the " geometrical co-ordinates involve the time."
ENSEMBLES OP SYSTEMS. I 239
tions as to continuity and finiteness are imposed on the
functions involved restrictions which we believe are obeyed
in these applications. The place where the ice is thin is the
implicit assumption that, for example, </> (x \ $x, y) will
practically serve as the expression for the velocity at any
point on the left-hand side Sy of the rectangle.
Gathering together results (23 . 1 . 6), (23 . 1 . 7) and (23 .1.8),
we obtain for the net loss of R-points in time from the
rectangle
/^. <~\
% ;r- !/ (x, V, #(*, V) j ZxSt + dx~\ f(x, y, I) ^(x, y) \ 8y 8t
dx ' J cy ' '
or briefly
+ ^SxSyS* . . (23.1.9)
But after time 8t the density of the R-points at the point
(x, y} becomes
f(*,y,t) + ^ t f(x, y, t) St.
Thus the net gain of R-points in the rectangle is
gfly St 8 X 8y,
or ^ 81 8x By . . . . (23 .1.10)
Ol>
Since results (23 .1.9) and (23 .1.10) must be consistent, it
follows that
=
dt Sx dy
Another simple step leads to
dt ox cy \dx dy
This is the " equatiqn of continuity," which must be satis-
fied since raindrops are not being created or destroyed. But
as u and v satisfy (23 .1.5), we conclude that
I' + ^-f^o . . . (23.1.12)
'
240 STATISTICAL MECHANICS FOR STUDENTS
This is the result of combining continuity with dynamical
law.
After time 8t, the R-point which was at the point (x, y),
has arrived at the point (x -f- u St, y + v 8t), and the
density of the R-points around it is
f(x + u 8t, y + v 8, t + 8t)
i.e.,
or
which by (23. 1 . 12) is simply p. Thus as the cloud of R-
points moves about in the phase-diagram, in a manner
entirely determined by the dynamical behaviour of each
member of the ensemble of systems, the density around any
given R-point (i.e., representative of a given drop of the
ensemble), remains unchanged. Looked at in another way,
this means that each element of area containing a definite
number of R-points does not vary in size, and so this must
also be true for finite stretches of the phase-diagram. The
reader is warned that the proof does not concern the density
around any given point of the phase-diagram ; i.e., it is not
proved that f(x, y, t + 8t) is the same as f(x, y, t), or that
dp/dt is zero. Put it in the language of Gibbs, we prove
that "in an ensemble of mechanical systems identical in
nature and subject to forces determined by identical laws,
but distributed in phase in any continuous manner, the
density -in -phase is constant in time for the varying
phases of a moving system ; provided that the forces of a
system are functions of its co-ordinates." In this the
emphasis is on the phrase " varying phases of a moving
system " ; the proposition is not necessarily true for the
density around a constant phase.
To be sure our proof so far has been limited to systems
with one degree of freedom ; then we can appeal to our
powers of visualisation to assist the understanding. That
ENSEMBLES OF SYSTEMS. I 241
the proof is general enough to cover any system with one
degree of freedom (and not merely the falling particle)
provided the force is a function of the co-ordinate, requires a
very little adaptation at the beginning, which will be appre-
ciated presently when we take the case of more complex
systems. But before passing on it will be instructive to
consider another simple system, viz., the pendulum, since it
illustrates very clearly a point which is of prime importance
in the statistical applications.
In the ensemble of pendula all members have identical
lengths, and swing in places with a definite value of g.
In fact I and g are parameters. But the amplitude is an
arbitrary constant introduced in integration of the equations
of motion, and varies from member to member of the
ensemble. The second arbitrary constant is the " epoch-
angle " which settles at what instants the string is in a
definite position, say the vertical. Further and this is
important the periodic time is not the same for different
amplitudes. The elementary theory of the pendulum,
which brings out the value 2?r (l/g)* for this, considers only
infinitesimal swings. The true result is 2?r (Ifg)* /(a), where
/(a) is a function of the amplitude a concerning which all
we need to specify at this moment is that it approaches
unity as a approaches zero, but increases in value with a
increasing. The angular co-ordinate, 0, and the angular
momentum ml 2 will be represented in the phase -diagram.
The R-point of any system will travel round a closed oval
curve with as centre these ovals will approximate to
elliptical form for small axes but the time of making a
complete circuit increases somewhat as the ovals increase
in size. Take two points, P, Q, on one of these ovals and
draw normals at them, cutting another oval outside and
near it in points P', Q'. The figure P Q Q' P' is quasi-
rectangular. Conceive a cloud of R-points distributed
uniformly throughout it. As the corresponding systems
in the ensemble oscillate in accordance with dynamical law,
the R-points travel round the appropriate ovals. All those
on one oval go round in the same time, but those on a given
oval take a little longer than those on one within it. Thus
242 STATISTICAL MECHANICS FOR STUDENTS
after one period for those on PQ, those originally on P'Q'
have not quite arrived back, and similarly for those between
PQ and P'Q' ; there is a progressive lag as we go outwards.
So the shape of " cloud-covered " patch is gradually more
and more distorted from the initial " quasi-rectangularity "
into a quasi-parallelogram shape, but the area remains
unchanged. After some time the inner R-points will have
overtaken the outer, and so the area is stretched like a
spiral ribbon round between the inner and outer ovals.
Still later we can have a very thin ribbon indeed spiralling
many times round as we go along it from inner to outer
ends. Actually if the R-points were visualised as black
dots on white paper, the whole space between the two ovals
would present the appearance of blackness all over, or rather
a continuous grey ; and so if P Q Q' P' were an area equal
to l/n of the ring between the ovals it might appear that the
R-points had now a uniform density equal to l/n of the
original ; yet this is quite wrong in reality. Density,
remember, is regarded as a function of x, y, t ; it is obtained
by dividing a number of R-points in an element of area by
the value of that area and proceeding to a limit. If we
conceive this clement of area to be in motion with the
central R-point of it, we preserve unvarying density. If we
conceive the element to be fixed, it is alternately empty
and full of points. In other words, as time goes on the
" black ribbon " grows longer and thinner, but the " white
ribbon " in between does the same and maintains its size
also. The sizes of the two parts maintain the same ratio,
but of course the R-points are not so compactly situated
as at the beginning, giving the impression of a uniformity
of distribution over the whole annulus between the ovals
which is an illusion. This reference to such an ensemble
will help us in certain statistical considerations which will
arise shortly.
23 . 2 Systems with Two Degrees of Freedom. As the
simplest illustration of this type, let us consider a particle
moving in a plane in a field of force due to a centre of
attraction or repulsion at an origin, and choose polar co-
ordinates r and 6. Anyone wit)) some knowledge of the
ENSEMBLES OF SYSTEMS. I 243
dynamics of a particle knows that the equations of motion
are
m (r - r9 2 ) = F
m | (M) =
at
where F is the force directed from the centre. Now the
kinetic energy is | m(r 2 + r 2 2 ) and the potential energy
V(r) where F = dVjdr. The total energy is
The partial differential coefficients of this with regard to
r and 6 respectively are mr and wr 2 ; these are the
" momenta " in the general sense used throughout this
book they happen to be the ordinary linear momentum of
the particle resolved along r, and the angular momentum
round the centre. Call them p r and p d . It easily follows
that the energy is equal to
(23.2.1)
and it requires no great trouble to establish that the equa-
tions of motion above are equivalent to
dp r == _ dE(r, fl, p r , p e )
dt Sr . . (23 . 2 . 2)
dt 38
where E(r, 6, p r , p e ) is written for (23 .2.1).
It can also be seen without much effort that
dr _ 8E(r, 0,p r ,p e )
dt 8 ^ ... (23.2.3)
dO
dt dp d
* It happens that the function E does not contain explicitly, and so
9E/30 is zero, but this accident does not invalidate the generality of the
reasoning.
R 2
244 STATISTICAL MECHANICS FOR STUDENTS
In fact, (23.2.2) and (23.2.3) constitute Hamilton's
form for the equations of motion.
Conceive now the possibility of a representation in a four-
dimensional phase-diagram, with r and 6 represented along
axes OX 1? OX 2 and p r and p e along axes OY 1? OY 2 , so that
if we denote the components of velocity of an R-point,
representing some system, along its path in the diagram by
u v u 2 , v v v 2 , where u l = r, u 2 = 0, v^ = p r , v 2 = p e we
have
9 x 2 , y v y 2 )
j, x 29 y l9 y 2 )
dx l
9 x 29 y l9 y 2 )
dx 2
From this it follows that
which corresponds to (23 .1.5) and leads to similar con-
clusions ; but more of this presently.
23 . 3 A Rigid Body. This will briefly illustrate a system
with six degrees of freedom. Three of these are easily
disposed of ; they correspond to the three ordinary co-
ordinates of a point fixed in the body, say its centre of mass.
The other three are a little troublesome to visualise ; the
reader will be helped by considering the earth in relation
to the plane of the ecliptic and the fixed stars. These
latter give us a " system of reference/' viz. 9 the plane and
some definite direction in it, say from sun to a definite point
in one of the constellations of the Zodiac. As regards the
earth we choose a convenient plane in it, and some definite
direction in that plane, say the plane of the equator and a
line from the centre to the point on the equator with longi-
tude zero. Now the plane of the equator at any moment
ENSEMBLES OF SYSTEMS. I 245
intersects the plane of the ecliptic in a line, viz., the " nodal "
line, and there is a definite angle between the planes, the
" inclination." This angle constitutes one co-ordinate ;
in the case of the earth it remains nearly constant in time,
only experiencing a secular periodic variation the " nuta-
tion " of the earth's axis. Then the nodal line makes with
the line of reference to the constellation point (they both
lie in the plane of the ecliptic) a definite angle, which is a
second co-ordinate ; in the case of the earth there is a slow
increase of this angle at the rate of about one revolution in
27,000 years the " precession " of the earth's axis. Lastly,
the nodal line makes a definite angle with the line on the
earth from centre to zero longitude on the equator. In
the case of the earth this third co-ordinate varies by 360 in
one sidereal day. This method of choosing the " Eulerian
angles " can be applied to any rigid body in a given frame
of reference. Works on dynamics show how the kinetic
energy can be expressed as a homogeneous quadratic
function of i, y, x, 9, </>, *jj, where 6, <f>, ifj are these
angles. One important feature is that the coefficients of the
individual terms may involve functions of the angles, such
as cosQ, sinO, sin<f>, etc. ; but we noticed that in the previous
example a coefficient of a squared velocity term might
involve a function of the co-ordinates (e.g., mr 2 d 2 ). Another
feature is that product terms in the velocities as well as
squared terms make their appearance.
CHAPTER XXIV
ENSEMBLES OF SYSTEMS. II
24 . 1 Hamilton's Equations. In considering more com-
plex systems than in the previous chapter, we can realise
that the complete configuration of a system will be given
by a suitable number of co-ordinates. Three of these would
specify the position in a reference frame of some definite point
of the system, say its centre of mass. For each particle
in the system not rigidly bound to other particles we would
have three further co-ordinates in a frame of reference
having this point as origin. If any rigid bodies entered
into the constitution of the system, each of these would
introduce six co-ordinates. These co-ordinates (in most
practical applications they are Cartesian or polar co-
ordinates, with Eulerian angles in the case of rigid bodies) are
denoted usually by symbols such as q l9 q 2 , . . . , q^ where j8
is the number just necessary and sufficient for complete
description of the configuration of the system. The Car-
tesian co-ordinates of every particle or element of volume
of the system can be expressed as functions of q v # 2 , . . . ,
ft; e '9->
XT = <l>r til, ? ft)
Vr = tr til, V* ft) *
*r = X r til, ?2> ft)-
The velocity of each particle or element of volume is given by
d(f> r . dJ> r . d6 r .
^ = ~q^ + ^T-<l2 + + 3^,
dq l dq 2 dq ft *
and two similar equations.
Recalling the expression for kinetic energy, viz.,
-Zm r (x* + 2/r 2 + 2> 2 )> it is clear that the kinetic energy of
2*
* In certain special cases t is also involved in the functional form,
i.e., the geometrical equations involve the time explicitly.
246
ENSEMBLES OF SYSTEMS. II 247
the system is given by a homogeneous quadratic function of
the generalised components of velocity q i} q 2 , . . . qp, the co-
efficients of the squared and product terms being in general,
not constants, but functions of the co-ordinates. Denote
this by
2 ( a n tfj 2 + + 003 <lf? + 2 12 qi q. 2 + . . .) . (24 . 1 . 1)
The generalised components of momentum, denoted by
Pii P& - P/3> are obtained by differentiating this expres-
sion with respect to q l9 jo, . . . , j^, respectively. Thus
Pr = r 1 ?1 + r2 </2 + - - - + ^ ft
where it is implied that a rs = a sr . It is now possible to
express the kinetic energy in terms of the components of
momentum. It is also a homogeneous quadratic function
of these, such as
\(b llPl * + . . .+b ftft p f * + 2bup l p + . ..) (24.1.2)
where the coefficients b rg are in general functions of the
co-ordinates.
The potential energy of the system, whether arising from
inter-action between its parts or from action between these
and external bodies is a function of the co-ordinates V^,
?2 ft)> or briefly V(^), involving among its parameters
any quantities necessary for specifying the relations of the
system to such external bodies. The sum of this function
and the function expressed in (24 .1.2) is called the
" Hamiltonian function " of the system. Let us denote it by
E (?i, ? 2 ?0> Pi P^ Pp), or briefly E (?, p) ; its
value for given values of the q r and p r is, of course, equal to
the energy of the system when in that phase of configuration
and motion. The importance of this function lies in the
fact that nearly a century ago the Irish mathematician,
Sir William Rowan Hamilton, demonstrated that the equa-
tions of motion of the system could be written in the form
dq, = 9E (q, p)
dt * P > ... (24.1.3)
dp, = __8E(g,y) V ;
dt d
248 STATISTICAL MECHANICS FOR STUDENTS
there being /? equations of the first type, and p of the second.
Even in the case of non-conservative systems where the
forces cannot all be derived from a potential function,
provided such forces are functions of the co-ordinates, we
can write the equations of motion
dq r _SE(q,p)
dt **> . (24.1.4)
where the ft functions Q, (q) of the co-ordinates depend on
the forces which are not included in the potential function
V (q). Representing dq r /dt, or q r , and dpjdt, or p, , by u r and
v r respectively, we see that each of the u r and v r are functions
of the phase quantities q lt g 2 , . . . q ft9 p l9 p 2) . . . p ft .
Further, it follows from (24 . 1 . 3) or (24 . 1 . 4) that
dq r dq,
dp, dp, Sq,
Hence we obtain
. . .(24.1.5).
dq r dp r
24 . 2 Liouville's Theorem. We can now prove the general
theorem mentioned in the last chapter for an ensemble of
systems all identical in nature but of any degree of com-
plexity. Each system of the ensemble has its own individual
history, but the initial conditions of each system have been
so chosen that at time t, the phase of any system is different
to that of any other. Yet the number of systems in the
ensemble is so great that within a finite extension-in-phase
there are an enormous number of them. An element of
this extension-in-phase can be specified by the limiting
values
ft + S ?2>#i + 8 ^i" "Pf* +
ENSEMBLES OF SYSTEMS. II 249
meaning that a phase in this extension is given by q l
a v . . . q ft dp Pib v . . . p ft bp where a l9 . . . b ft
are 2 /J small positive quantities less in numerical value than
i 8j lf . . . ^ Spp respectively. The number of systems of
the ensemble within this element at time t is
where p is a function of the q r and p r and Z, say / (q, p, t).
To find the number at time t -f- 8t we must calculate the net
gain of systems within these limits in time t, due to the fact
that the phase of each system has altered in the interval.
If 8t is chosen small enough, systems which have at time t
phases given by
0i- 5 8 ?i - *n 02 2> 9 ft *v Pib v . . . Pfi b ft ,
u
and which are therefore outside the element at time t will be
within the element at time t + 8, provided h is not greater
than (u l | dujdq^ SqJ St. On the other hand those
whose phases at time t are given by
?1 +-3 ^ ~ k l> ?2 2 ' ' & ft /3' JPl 6 1 ' Pft V
and which are within the element at time 2 will be outside it
at time t + S provided k^ is not greater than (u l + 9^ 1 /3g 1 .
Sg^) 8^. The net loss of systems to this element in the
interval S on this account will be given by
~ Ul 8 ?i & 8g 2 ... 8^ 8p l . . . 8p ft .
C$1
Proceeding in this way the complete net loss in the
interval Stf is given by
S* 8q l . . . 8^ 8^! ... Sjpj X
But the net gain is also
? 8* S .
250 STATISTICAL MECHANICS FOR STUDENTS
Hence, since these expressions must be consistent we have
the fundamental result
...... _ Q ^ (24.2.1)
dt dq dp ft
But by reason of (24 . 1 . 5) this reduces to
. . (24.2.2)
.
ot cq r
Now the system of the ensemble which was at the phase
<?!> Pp at the time t is at the phase q l + u 8t, . . .
Pp + Vp St at time t + 8t and the density-in-phase of the
ensemble at this phase is f(q + uSt, p + vSt, t + or
which, on account of the condition (24 .2.2), is just p.
This establishes the general theorem referred to in the
previous chapter. Expressed in other words this means
that if a particular system of the ensemble is at a phase
?i,... pp at time t and p is the limit of the number of
systems within an element of cxtensioii-in-phase q l i a l9
- Pp bp a ^ that time divided by 2^ a l . . . a^ 2& b l . . .
6^ when a 1? . . . b^ are indefinitely reduced, and if the
particular system is at phase q' l9 . . . p'^ at time t' and p
is a like limit at time t', then p = p' . The use of the methods
of the calculus imply that we must conceive the number of
systems in a finite extension -in -phase to be increased without
limit, just as in hydrodynamics we conceive a fluid to be a
continuous medium. There is still another way of looking
at this important proposition. We can assume that all
the phases in a certain finite extension-in-phase satisfy the
condition that each of them make a function (f> (q> p) of
?!,... Pp negative or zero in value. Using geometrical
language we can say that those phases which make the
function zero are on the " hypersurface," </> (q, p) = 0, and
the others " within " it. Suppose a number of systems of
the ensemble are at these phases at time t. At time t' they
are at other phases, and from the dynamical equations
?i'> P'ft can be expressed as functions of q ft , . . .
ENSEMBLES OF SYSTEMS. II 251
p ft and ' t, so that we can obtain a function i/j (q', p')
of q' l9 . . . p'p which is equal to < (q p). Then all the
systems at time t' will be within the hypersurface
ifj (q, p) - 0,
or on it. Liouville's theorem then states that the integra-
tion of dq . . . dq ft dp l . . . dp ft throughout the extension
determined by </> (q, p) being zero or negative yields an
integral equal in value to that obtained by integrating
dq t . . . dq^ dp-^ . . . dp^ throughout the extension deter-
mined by i/j (q, p) being zero or negative. Thus there is a
conservation of the extension-in-phase occupied by a group
of systems distributed continuously in a finite extension to
begin with.
One word of caution is necessary. The actual systems
which lie within the " rectangular " element q l a l9 . . .
Pp it ft/3 at time t, are not in general the same as those which
lie within the rectangular element q\ a v . . . p' ft fys
at time t f . The number is the same, but identity is not
necessarily preserved. Put in geometrical language, the
" shape " of <f> (q, p) ~ is not necessarily the same as
that of (q, p) = 0. If a rigid body moves in our per-
ceptual space, certain relations must hold between the
co-ordinates x, y, z of a particle of it at time t, and the
co-ordinates #', y', z f of the same particle at time t' by reason
of the fact that the moving thing is rigid and preserves its
shape. But no analogous relations hold between q l9 . . . p ft
and q' l9 . . . p'^ ; at all events Liouville's theorem makes
no such claim ; it simply proves conservation of the exten-
sion.
Last of all we have clearly not proved that dp/dt is zero,
or that the density-in-phase about a particular phase is
conserved.
It is, of course, understood in all this that no system of an
ensemble acts on any other system ; each pursues its pre-
destined dynamical " path," under the influence of its own
internal interactions or of the actions of external bodies,
and it should be carefully noted that the disposition of
external bodies is supposed to be alike for each system.
252 STATISTICAL MECHANICS FOR STUDENTS
Each member of the ensemble is really system plus external
bodies. At a given instant the phase-quantities which deter-
mine the configuration and motion of the various parts of
a system differ from the member to member of the ensemble,
but those parameters which determine configuration and
motion of the external bodies are alike for all members.
These parameters may, of course, change in time, but at a
given instant they have identical values for all the members.
As an example of this we have the ensemble of rain-drops or
pendula treated in the last chapter. Another example, an
ensemble of gas systems, is treated on the assumption that
each one is enclosed in one of an enormous group of identical
vessels, i.e., each one is subject to an external field of force
which is alike for all.
24 . 3 Microscopic States and Their Probabilities. A
" microscopic state " of a system is defined by stating that
the phase of the system lies within a small rectangular
element of extension-in-phase about a particular central
phase.
Now the successive phases of an actual system must, by
the laws of dynamics, satisfy the relation
E (q, p) = constant.
These are, in fact, in geometrical language, the energy
hypersurfaces, and the dynamical path lies on one of them.
It is clear, therefore, that the phases possible to a system
with a given energy do not occupy an extension with the
full dimensionality, 2/?, of the ensemble. We can, however,
conceive an ensemble of systems limited in such a way that
no system has an energy less than a definite value X, and
none has a value more than a value X -f 8X, greater by an
infinitesimal amount. Imagine also that at any instant
the systems of the ensemble are so distributed in phase
that their density-in-phase is uniform throughout the
extension bounded by the hypersurfaces.
-E(q,p)=X
E (ff, p) = X + 8X . . . (24 . 3 . 1).
Liouville's theorem then shows that this state of things is
ENSEMBLES OF SYSTEMS. II 253
perpetual. In fact, since at time t, the 9p/3g f and
are all zero, it follows that
also. Thus the ensemble is in statistical equilibrium.
There is no tendency for the systems to crowd into any
particular part of the extension defined by (24 . 3 . 1) at
the expense of other parts.
Imagine that we are endowed with the power of making
a choice of one system out of the ensemble. We are just
situated in a position similar to a person asked to select
any ball out of a number of boxes each containing the same
number of balls. The chance that the choice will fall
within a certain box is I/A if there are A boxes. So the
chance that we select a system within a given microscopic
state is I/A if the extension defined by (24 . 3 . 1) is divided
into A equal elementary extensions. Or to put it another
way, all the different microscopic states of the ensemble
have equal probabilities. Further, this chance is not
upset with lapse of time, as would have been the case had
Liouville's theorem not been true, for then systems would
have crowded in increased numbers into certain regions of
the complete extension, giving, therefore, after a time, an
enhanced probability that one of the microscopic states
within this region would contain the system chosen, the
chances elsewhere being on the other hand diminished.
An actuary, it is well known, when preparing tables for
insurance or other purposes does not base his calculations
on the facts concerning one or two individuals, but on
averages extending over large groups representative of each
section of a community of individuals. Our point of view
at this stage is very like that of the actuary. We really
have not been basing our statistical calculations in the
preceding chapters on the behaviour of a single molecular
system, but on the average behaviour of a large representa-
tive group of systems, members of an ensemble. Each
system is composed of the same number n of molecules, all
of the same kind, each with / degrees of freedom. If more
254 STATISTICAL MECHANICS FOR STUDENTS
than one type is present, each system contains n molecules
of one type each witH / degrees of freedom, ri of a second
type each with /' degrees, and so on. The number of
degrees of freedom of the system is nf, or nf + n' /' -J-, etc.,
as the case m&y be. Thus in the case of one type of molecule
^ #ii> #12 -j (?i/ denote the co-ordinates of the first
molecule in the system q 2l , # 22 , . . ., q 2f those of the second
molecule, and so on, then the co-ordinates of the system
are q lv , q nf , the number being nfin all; in short,
the suffix j8 of the preceding portion of this chapter is nf
(or nf + n' f + etc.). It is very necessary to be on guard
here against a fatal misconception. In a sense a molecular
system is an ensemble, for it contains an enormous number
of molecules, and each molecule is in itself a dynamical
system. But that is not the sense in which the word
ensemble is used here. For one thing, the molecules inter-
act on one another, and Liouville's theorem certainly does
not hold for a molecular system regarded as an ensemble of
molecules. The systems must be regarded as independent
of each other, each pursuing its own dynamical path un-
interfered with by any other system if Liouville's theorem is
to be true. It is as well to introduce here a simple notation
to indicate the distinction made. We have in the past
spoken of representing the momenta and co-ordinates of a
molecule in a phase-diagram or " phase-space " with
dimensions 2/ ; we shall refer to this as the M-space. But
now we must also think of a phase -space in which a whole
molecular system is represented by a single " point." Such
a space has a dimensionality 2/3, where /? = nf (or nf +
ft'/' + etc.). We shall call this the G-space. (Gibbs'
phase-space.) It is in this space that an ensemble of
systems is represented by a " cloud of points."
A microscopic state of the system is obtained by assigning
narrow limits to the /3 co-ordinates and /? momenta of the
system. A moment's thought will show that this is tanta-
mount to assigning a complexion to the molecules of a
molecular system. Thus if all the systems are considered
to have values of energy practically the same for each, i.e.,
between narrow limits such as X and X -f SX, then each
ENSEMBLES OF SYSTEMS. II 255
complexion of the system, which is indicated by assigning
n l particular molecules to the first phase-cell in the M-space,
n z to the second, . . ., n c to the c th , has, as it were, one
particular phase-cell in the G-space within the shell between
the two hypersurfaces as its home. A statistical state of
the system is therefore associated with a particular group
of phase-cells in the G-space, the number in the group
being n\/(n- L \n 2 \ . . ., n c !). But we have seen that all
but a relatively insignificant number of complexions are
embraced within a statistical state in which the number of
molecules in an M-phase-cell, corresponding to molecular
energy , is proportional to e~^ e (where 3/2/x is the average
kinetic energy of translation of a molecule) or agrees very
closely to this distribution. This means that if we fill the
shell in the G-space with a uniformly dense cloud of repre-
sentative points, then on choosing one at random there is an
enormous probability that we shall select one which repre-
sents a system in, or very near to the Maxwell-Boltzmann
distribution of molecular co-ordinates and momenta.
Moreover, and this is where Liouville's theorem comes in,
this state of affairs if arranged for initially is not upset in
course of time. The distribution in the G-space remains
uniform, and so on, returning to make a choice at a later
instant the chances are still enormously in favour of select-
ing a representative point associated with a system in or
near the Maxwell-Boltzmann distribution. In other words,
this distribution is a normal property of the system. The
word " normal " was suggested by Jeans as a convenient
epithet for any property which is common or nearly common
to every member of the ensemble distributed in a uniform
manner throughout a region of the G-space. As giving
point to the warning uttered above, the reader will observe
that the distribution of representative points in a uniform
manner in the G-space does not lead to uniform distribution
of M-representative points in the M-space, except in rela-
tively few cases. An enormous preponderance lies with a
very non-uniform distribution in the latter space.
This is essentially an actuarial process, and actuaries
know that, for the purposes involved, although the life of a
256 STATISTICAL MECHANICS FOR STUDENTS
given individual may in its course be seriously at variance
with the average life of the whole community, yet things
will work out alright in the end on the assumption that it
is the same in certain particulars. This is essentially the
hypothesis we introduce at this point of our reasoning.
Dynamics can carry us no further. It has had its say.
A given molecular system might behave very differently
from one which is always in or near the Maxwell-Boltzmann
distribution. There do exist relatively very small regions
of extension-in-phase in the G-space where matters are very
different for any system whose representative point happens
to be there, and we cannot definitely deny the statement
that once there it will always remain in such a region, or
that the greater part of its path may lie in such regions.
All we can say is that it is unlikely. But the reader must
carefully note that in carrying over what is undoubtedly a
proven statement for the average behaviour of all the
systems in an ensemble at any instant, for application to a
single system, and asserting that such is the average be-
haviour of a single system throughout a long time, we are
taking a step which is undoubtedly plausible, but which
cannot be definitely proved. The hypothesis was called by
Maxwell the " principle of continuity of path " and by
Boltzmann the " crgodic hypothesis/' It amounts to an
assumption that a given molecular system will in course of
time pass through all the complexions consistent with its
energy, or at all events through a large group of them
sufficiently representative to them all, before returning to
or very near to some original phase, and thereafter pursuing
the same path as before. Yet it must be admitted that
the dynamical systems which have been most thoroughly
worked out in detail by the mathematicians, viz., the astro-
nomical systems, give little or no support to this view.
Indeed we can realise that our own solar system, if absolutely
free from all external influence and left to the internal
gravitational action of its parts, would go on cycle after
cycle, much as it is, never approaching in successive ages
unlimited phases which we can conceive it to have by tum-
bling planets and orbits about in our imagination. There is,
ENSEMBLES OF SYSTEMS. II 257
of course, an intuitive feeling that the very complexity of
molecular systems and the multiplicity of their parts favour
the hypothesis, but this feeling does not constitute
proof. Perhaps the most helpful idea is contained in the
undoubted facts that any molecular system can hardly, in
practice, be said to be free from other influences than those
ostensibly introduced in theoretical discussion. For example,
even in a thermostat slight fluctuations of energy must go
on in any system ; this means that though for a time the
system may pursue a definite path on one energy hyper-
surface, presently it will be " displaced " to another on a
neighbouring hypersurface ; and although any one path
might be far from traversing a sufficiently representative
group of microscopic states, yet the fortuitous shifting
from path to path may produce this result in the long run,
and effectively prevent any system from remaining in a
freak region of extension-in-phase for anything but a
negligible period.
The " freak " regions are relatively very small, and any
system might be compared to a man blindfolded and left
to wander at random in a large field. The chance that he
would walk into a small circle drawn in a particular spot on
this field is small. If in the circle, the chance that he would
remain in it and not wander out of it is also small. There is
no impossibility in his walking into the circle and " depress-
ing his entropy," but in all likelihood his entropy will soon
rise again, and he will walk out of it.
Let the reader also recall the example given in the previous
chapter, where an ensemble of simple systems, compactly
grouped in the beginning, " spread themselves out" with-
out contradicting the law of density-conservation ; so
that a choice at the beginning would have given one of a
rather restricted group of phases, while at a later time a
choice would be made from a group of phases more repre-
sentative of all the phases possible. The reader should in
this connection consult Gibbs' Elementary Principles,
Chapter XII.
The ensemble method is very powerful in dealing directly
with problems where we are concerned with a system which
258 STATISTICAL MECHANICS FOR STUDENTS
is not isolated, and it is a question of the probability that it
contains a certain energy, or rather that its energy is within
certain narrow limits. This probability is proportional to
the number of microscopic states consistent with these
energy limits, i.e., to the magnitude of the extension-in-phase
bounded by the energy-hypersurfaces corresponding to the
upper and lower limits. Of course, when quantum assump-
tions come in we have to modify this strictly classical
result in a manner which will be obvious on glancing back
at Chapter xvi. A microscopic state surrounding a
particular phase is the element in the classical discussion ;
in the quantum discussion the element is the whole succes-
sion of phases in a given quantum state. The probability
in the one case is the magnitude of the G-phase-cell of
dimensionality 2/3, i.e., having physical dimensions equal to
action raised to the power /?. In the other it is h ft . To
illustrate this procedure we shall return to a fresh discussion
of the entropy-constant problem, but before doing so we
must point out the importance of choosing momentum in-
stead of velocity to indicate phase. We have indulged in this
practice throughout, and the cause is clear. The elegant
form of Hamilton's equations, leading as they do to the
fundamental statistical theorem on density -conservation,
depend entirely on choosing the p r , and not the q r , as
indicating the motion aspect of the phase. In a phase -
diagram in which q r and q r are represented there is no
conservation of density of representative points around a
moving phase, and no simple basis for probability calcula-
tion presents itself.
24 . 4 The Entropy-Constant Once More. Let us consider
once more a system of n simple monatomic molecules
represented by a point in a G-space of 6n dimensions. Sup-
pose the system to be in a state in which a particular atoms
are in the gaseous state, and the remaining b particular
atoms are in a solid cubic lattice, and arranged in a particular
way in the lattice.
To simplify the discussion we will also assume that the
lattice is also in its lowest quantum state, so that its energy
is be where is a constant. In order that any molecule
ENSEMBLES OF SYSTEMS. II 259
may leave the lattice and enter the gaseous state its potential
energy must be increased by an amount w. Hence, if
K is the kinetic energy of the gaseous molecules, then
K + aw + 6e must lie between x and X + ^X' Regarding
for a moment the gas molecules as a separate system with
3a degrees of freedom, the probability that it will contain
an amount of energy whose value lies between -^ aw be
and x aw be + <>x * s proportional to the magnitude of
the extension in a Ga-dimcnsional phase-space bounded by
the energy hypersurfaces
(^ + ^ + i 2 + + a + ^ + C 2 )
= 2 m (^ aw be),
and
= 2 ra(x aw be
It will be shown in a note at the end of the chapter that
this is
(277-m) c v a { aw _ & y-i g (24.4.1)
(c - 1) ! V * ' * V '
where v is the volume of the gas, practically of the containing
vessel, and c = 3a/2, a being assumed to be an even number.
Since we have limited the lattice to one quantum state,
viz., the lowest consistent with the number of molecules in
it, the probability that it is in this state is given by an
extension in a 66 dimensional phase-space. This is h* b
according to the quantum postulate just referred to. Thus
the probability that the a particular molecules are in the
gas and the remaining b in the lattice arranged in a particular
way and in their lowest quantum state is given by an
extension in a Pm-dimensional phase-diagram, whose magni-
tude is
h*(2*mY( X --b*Y- 1 8v / 24 4 2]
(c 1) ! A v ' ' '*
The probability that any a molecules may be in the
gaseous state, and the remaining 6, still arranged in one
way in the lowest quantum state, in the solid is obtained by
multiplying (24 . 4 . 2) by n \/(a ! b !), which is the number of
82
260 STATISTICAL MECHANICS FOR STUDENTS
ways of choosing a molecules out of n. But this has still to
be multiplied by b \ to introduce all those complexions
which correspond to any arrangement of the b molecules in
the lattice. Thus the probability that there are a molecules
in the gaseous state out of the total number ?i, is
_ - h v a (2rr m) c (v - aw - be)*" 1 Sy . (24 . 4 . 3).
a!(c-l)! * *
The most probable value of a is found by finding a maxi-
mum value of this expression, subject to the condition that
^ and 8^ are constant. Referring to the expression
(24 . 4 . 3), with n ! and 8^ omitted, as P, we, as usual,
take logarithms and use Stirling's theorem. Thus,
3#
log P = 3 (n a) log h -f- a log v + IT 1 (^ m )
If we now write down the condition that d log F/da is
zero we find, after a little re-arrangement, that a is deter-
mined by the equation
5 3 3 e w
-log a - log fv ne 4- a (e iv)\ -- - - ; - --- :
2 fe 2 6LA - v yj 2 ^~H + a (~ w)
3 3 3
= - 3 log h + log v + 2 log 277 m - ^ log--
If we now connect this statistically most probable dis-
tribution with the macroscopic experimental state, for which
the given total energy ^ determines, with the volume v and
the number a, the temperature 0, we know that ^ aw
6e, being the kinetic energy of the gaseous molecules, is
(3/2) a k ; further we know that the pressure p of the gas
is a k 6/v, since v is practically the volume of the gas. Hence,
since log p = log a log v + log k9 we readily find that
3 , 7 n e w
Ioga--logk9--^j-
3
= 3 log h + log v + - log (2?7 m)
* |n practice a is so large that we can replace 3a/2 1 by 3a/2,
ENSEMBLES OF SYSTEMS. II 261
w e 5 3
or logp = - + + g log (0) + - log (277 m) - 3 log h
w e 5 (277 m ft) k
This is the vapour-pressure equation for the simple model
we have chosen. It clearly corresponds to the general
thermodynamic equation (21 . 2 . 8) in Chapter xxi.
w 1 , L
-- - corresponds to -
ku R0
ice " "
5 S P
2 " " R
, (2rr m k) 3 - k
7 log fe8
The method which is due to Ehrenfest and Trkal is some-
what less laborious than Stern's, and differs from his most
signally in the fact that it treats the statistical considerations
on a quantum basis, while Stern's treats the statistics on
classical lines, while applying quantum methods to the
thermodynamical investigation.
24 . 5 Gibbs' Canonical Ensemble. In order that an
ensemble may form the actual basis for a statistical treat-
ment of a dynamical system, it is necessary that the ensemble
should be in statistical equilibrium. This means that the
density -in-phase of the ensemble round a given phase must
not change in time, i.e., that dp/dt must be zero. Now this
condition is not generally true. Liouville's theorem demon-
strates that the density around a dynamically-moving phase,
i.e., around the " point " representative of a given system,
is constant, i.e., that dp/dt + 2 q dp/dq r + 2 p r dp/dp r is
zero. It is therefore necessary for the statistical equilibrium
of the ensemble that the condition
= . . (24 . 5 . 1)
262 STATISTICAL MECHANICS FOR STUDENTS
be satisfied. If this is so, any assumption that the prob-
ability of a given microscopic state being occupied by a
single system at a chosen instant is proportional to the
density of ensemble at the central phase of that microscopic
state, will not be upset with lapse of time, as would be the
case if the conditions of density around given phases kept
altering.
The simplest type of ensemble satisfying (24 .5.1) is,
as we have seen, one uniformly distributed in phase, but it
is by no means the only one. If, for instance, the density
is chosen initially at each phase to be a function of the
energy associated with that phase, (24 .5.1) is satisfied
in cases where the energy of the system is conserved ; for if
p =
where ^ i g the value at the phase of the Hamiltonian
function E (g, p) of the system ; then
dq r d x ' dq r
and
dp r d x dp r
and so
dp , . 9\ _df( x ) ( dg
dq r Vr Z'pJ d x (dt~ dq r " dt ~dp r
.
d% dt
= 0.*
If we wish to maintain other suitable mathematical
conditions for the ensemble as well as that of statistical
equilibrium, the function f (\) must be subject to other
conditions . Thus / (^ ) should not be infinite anywhere . The
reader may think this condition strange in view of the fact
that we have several times postulated an increase of the
number of systems in the ensemble to an enormous value so as
* Note that this result is only true for conservation of energy. In other
cases the right-hand side is 2 Q r q r . See Hamilton's equations.
ENSEMBLES OF SYSTEMS. II 263
to produce conditions approximating to a " continuous fluid "
of representative points in the G-phase-space. But this is
not really the same thing. The function f (%) must n t
become merely an indeterminate infinity anywhere as it
would do if it were put equal to, say -%~ l or ^ an X> or so
forth. Writing it as N ^ (^), where there are N systems in
the ensemble, we see that
over the whole phase-diagram must be unity, and this
clearly debars us from the choice of many functions. Thus
</> (^) cannot be simply proportional to ^, nor can it have
the same constant value everywhere, although we may
give it a suitable constant value between a pair of energy
hypersurfaces and make it zero elsewhere, as, indeed, we
did in the preceding considerations.
Gibbs points out that an ensemble distributed in such a
way that the density-in-phase is at all phases proportional
to
exp f ^ ^M . . . (24 . 5 . 2)
where is a constant, which he calls the " modulus of
distribution," " seems to represent the most simple case con-
ceivable, since it has the property that when the system
consists of parts with separate energies, the laws of distribu-
tion-in-phase of the separate parts are of the same nature
a property which enormously simplifies the discussion, and
is the foundation of extremely important relations to
thermodynamics." Such an ensemble he names " canoni-
cal." An ensemble with a constant density between two
very near energy-hypersurfaces, and a density which is
zero elsewhere, he calls " microcanonical."
A considerable part of Gibbs' " Elementary Principles "
is devoted to working out the properties of canonical
ensembles and showing how the rational foundation of
thermodynamics is related to them. There is no need to
go into the matter very fully here. Gibbs' work is certainly
not easy reading for a beginner ; but the reader who has
264 STATISTICAL MECHANICS FOR STUDENTS
struggled through the previous pages to this point is no
longer a beginner. The author feels that he can be safely
left at this point to pursue his further studies in this classic
of Statistical-Mechanical theory, and will bring this book
to a close with a few remarks on the canonical ensemble.
In the first place one must carefully guard against con-
fusing the Maxwell-Boltzmann law of distribution of
molecular co-ordinates and momenta in a system of mole-
cules with Gibbs' specification of a canonical ensemble of
dynamical systems. On the assumption that the probability
of a statistical state of a molecular system is proportional to
the number of complexions consistent with it one can
deduce the result that the number of molecules in a given
element of phase-extension (i.e., M-phase) is proportional to
e~^ where e is the energy of a molecule at the central phase
in the most probable state. But this statement implies no
necessity that this is true all the time. There are un-
doubtedly fluctuations from this state, such fluctuations
being less marked the greater the number of molecules in
the system. Gibbs on the other hand considers ensembles
of systems. These systems are not necessarily molecules
nor even groups of molecules, although his work finds its
most ready application in dealing with systems of molecules.
His system is any group of objects subject to dynamical
law. Further, the systems of the ensemble are isolated
from one another. Whatever law of distribution is laid
down for them, they follow Liouville's law, and under
certain conditions this implies statistical equilibrium with
no fluctuation. The exp ( E/@) law satisfies these con-
ditions, and is laid down by Gibbs on grounds of mathe-
matical convenience, and because of its ready application to
practical problems, qualifications which are illustrated
freely in his book. Of course, if we choose to regard the
individual systems as huge molecules and imagine them to
be all immersed in an enormous body of gaseous fluid (a
" temperature -bath "), we could appeal to the Maxwell-
Boltzmann result and prove that the systems would, except
on rare occasions, be distributed in phase, as Gibbs laid
down for his canonical ensemble ; and this explains why in
ENSEMBLES OF SYSTEMS. II 265
recent literature, references to Gibbs' canonical ensemble
as a " temperature -bath " occasionally occur. Indeed, in
Chapters xx. to xxin., this point of view has been
actually adopted at certain parts of the argument.
Of course, the expression exp ( E/0) is multiplied by a
constant factor in the specification of a canonical system,
and Gibbs writes his rule in the form
... . (24.5.2)
where ^ is a constant for every system, just as @ is. In
fact, ^ is a function of & and the parameters of a system,
the parameters, as already stated, being any quantities
necessary to express unchanging metrical properties of a
system or its dynamical relations to external bodies which,
one must remember, are the same for all the systems at one
moment. is determined in terms of @ and the para-
meters a v . . . a e by the equation
all phases.
or
>j, / (* E
e~^= . . . e~i^dq l . . . dp ft . (24.5.3).
all phases.
Gibbs devotes several pages to the discussion of average
values in a canonical ensemble. Thus if u is a function of
co-ordinates, momenta and parameters, its average value
over the whole ensemble is given by the expression
q, . . . dp, . (24.5.4).
all phases.
As an illustration let us work out the average value of
3E
p r . where p r is any momentum. It is
dp r
3E
P ^ r
all phases.
266 STATISTICAL MECHANICS FOR STUDENTS
Now by (24 .5.3) thia is equal to
f* f 3E
J ' ' ' J Pr to
J ~" W*
f f
' ' '
J og J
. (24.5.5).
But by integration by parts
f *
Jj
Hence (24 .5.5) becomes simply 0. Thus the average
value of p r d~E/dp r for any momentum is the same for all.
Now E = Ep -f E ff where E^ is the kinetic energy and
E ? is the potential. E ? is independent of the momenta,
and Ep is a homogeneous quadratic function of the momenta.
So
I
2 r =l Pr ty f
I
This is Gibbs' version of the equipartition of energy
theorem. The kinetic energy of any system can be divided
into /J parts, and the average value of any part over the
whole ensemble is the same, viz., -| 0. Note that this
theorem is not found for a time -aver age of a part for a
single system, but for an average at one instant over all the
systems of the ensemble.
Finally we shall show one of Gibbs' thermodynamic
analogies. In equation (24 .5.3) let us vary the modulus
and parameters from the values 0, a v . . . a e to + 80,
a l + 8a v . . . a e + Sa e . We obtain
*
e~
ENSEMBLES OF SYSTEMS. II 267
or
exp
(24 . 5 . 6).
Now the expressions 9E/9a s represent the forces
exerted by the system on the external bodies ; for since E p
does not depend on the parameters which define the position
of the external bodies, and since the energy entering the
system through the action of external bodies is
z ^ &,.
,=i da s
ZSa.
s~\ca 8
it follows that the expressions 8E/3a 5 represent the forces
acting on the system due to external bodies. Writing A,
for 9E/3a g , i.e., the force exerted by the system on its
environment through the parameter a 8 , we can write
(24 . 5 . 6) in the form
8^^~ E 8fe> - 2 A, 80, . . (24.5.7)
where we recall (24 .5.4) and indicate an average value of
quantity over the ensemble by drawing a bar over it.
Writing <>(#, p, a) for the function (^P E)/@ and
4> (a) for (9 E)/@ we have
e __
8V = o 80 E A, 8a s
=i
and since
it follows that - * -
8E + 27 A, 8a 8 . . (24 . 5 . 8).
268 STATISTICAL MECHANICS FOR STUDENTS
The analogy with the entropy equation in thermodynamics
is apparent. <t> (a) is the analogue of entropy, E(a) of the
internal energy, 2A 8 8a s of the work done by a thermo-
dynamic system on its environment, & of the temperature,
M* (a) of the free energy.
NOTE ON THE INTEGRATION IN SECTION 24 . 4
If we integrate dg x d 2 throughout a region of plane space
defined by x 2 + f 2 2 < r 2 the result is Trr* 2 . We obtain
from this result the integral 'of d^ d 2 d 3 throughout a
region of space defined by ^ 2 -f- 2 2 + 3 2 <; r 2 by means of
the " zone " method. That is, we put it equal to
2 ' 77 (r 2 - x 2 ) dx
If we now write V n (r) for the result of integrating d^
dg 2 ...... dg n throughout a range of values defined by
i 2 + ^ 2 2 + - . . +&<**
we can obtain V w (r) from V ;i _ x (r) in an analogous manner.
Thus
f-
= 2r f V n _i (
Jo
r cos cos
where we write r sin < for x.
Putting V n (r) = A n r w , where A n is some numerical
multiplier depending on n, we have
The definite integral is known to have the value
1 . 3 . 5 . . . (n 1) 77 ., .
------------------- v ------- / __ ii n is even,
2 .4.6 ... n 2
ENSEMBLES OF SYSTEMS. II 269
and 2 . 4 . 6 . . . (n - 1) .. . , ,
i if n is odd.
1.3.5... n
So
A 1 . 3 . 5 . . . (n 1) TT A .. .
A n = 2 . A M T if n is even
2.4.6 ... n 2 n ~ 1
and
A , 2. 4.6.. .tt 1. ., . , ,
A n = 2 . A-., if n is odd.
1.3.5... n l
In either case A n = A n _ 2 ,
n
and thus since A 2 = TT and A 3 = - 7 , it is not hard to show
o
that
V n (r) = r n if w, is even
n-l
and - 2 - - - r n if n is odd.
1 . 3 . 5 ... ft
Taking w as even, the region within a " shell " bounded by
and f ! 2 + n 2 = ( r
is given by SV n (r), or
n
U 7T 2
2
(i)'
which is equal to
In the text r 2 is 2m (E aw 6e), and Sr 2 or 2r Sr is
2m SE, while n/2 is 3a/2 or c.
APPENDIX ON RECENT DEVELOPMENTS
QUITE recently, no later than 1924 in fact, an unexpected
turn was given to the statistical theory of physical systems
by the publication of a paper by Bose on Planck's radiation
formula. The novelty lay in Bose's modification of those
steps in statistical theory which are concerned with d priori
probability. Yet so far is it true to say there is " nothing
new under the sun " that we can find the germ of Bose's
idea in Planck's earliest papers on black body radiation,
and it will prove serviceable to refresh one's memory on the
discussion in section 14 . 2. In the traditional methods of
statistical theory each particle (molecule, atom, electron)
in a system is assumed to have a recognisable individuality,
and in this way one complexion is distinguished from another
even if they are embraced in the same state of numerical
distribution of the particles between the phase-cells. In
section 14 . 2, however, we considered complexions as settled
by the distribution of elements of energy among the particles,
and this led to the substitution of an expression such as
(n + c - 1) !
n ! (c - 1) !
for the value of a probability W instead of the usual
n \ n 2 ! . . . n c !
It is expressions of the former type which enter into
Bose's analysis and link it up with Planck's earlier treatment.
1. Bose's Statistics of Light Quanta in a Temperature-En-
closure. In his work Bose regards radiation as composed of
light-quanta, or particles with energy hv where v is the
frequency of the quantum. In accordance with present
views as to matter and energy, a particle is regarded as
having mass Jivjc 1 * and momentum hv/c where c is the
* I.e., mass at velocity c ; their " rest-mass " is zero, according to
Relativity theory.
270
APPENDIX ON RECENT DEVELOPMENTS 271
velocity of light. Each light-quantum is represented in a
six-dimensional phase-diagram representing position (x,
y, z) and momentum (f , 77, ). Of course
i 2 + i? 2 + e = (Av/c) 2 .
If we integrate dx dy dx d df] d throughout the region of
the phase -diagram in which the energy is not greater than
liv where v is a particular frequency we obtain
477
V -
3
where v is the volume of the enclosure containing the radia-
tion. In this region of phase-extension no points represent-
ing quanta of higher frequency than v can occur.
In the extension
will occur these points which represent quanta with fre-
quencies between v and v + Sv. If we now introduce the
assumption that the elementary phase-cells have the
magnitude h 3 , then the number of the cells in the shell
bounded by the frequencies v and v + v is
4 77 V
The problem is to determine the distribution of the num-
bers of light-quanta among the various frequencies in an
enclosure of volume v containing full radiation, i.e., radiation
in temperature equilibrium with the walls of the enclosure.
This condition determines the amount of energy in the
radiation, but it does not determine the total number of
light-quanta present. We cannot assign an unchanging
individuality to any light-quantum in the enclosure, since
the walls are emitting and absorbing radiation ; even the
number in the radiation is not necessarily unchangeable ;
for if a quantum of frequency v is absorbed by the wall it
can be replaced by emission of several quanta of frequencies,
v', v", . . . without any breach of the energy condition,
provided v is equal to the sum of v ', v" . . . . Conceive the
272 STATISTICAL MECHANICS FOR STUDENTS
whole range of frequencies to be divided into elementary
ranges between the frequencies v v i> 2 , . . . v r , etc. . . .
where we can write 8v r for v r v r _- L . Suppose the total
number of light quanta in the radiation at any moment to
be n, and that of these n^ lie in the range o to v l9 n% in the
range v l to v 2 , . . . n r in the range v r ^ l to v r , etc. We
have to find the number of complexions which corresponds
to this numerical distribution among the elementary ranges.
It is here that Bose resorts to the earlier method of Planck.
Take, for instance, the n r quanta in the range v r _ l to v r *
Their representative points lie in a shell of the phase-
diagram which is constituted of a r elementary phase-cells
where
/T ,.
Vr r ' ' ' ' ( ^
As we have seen in section 14.2 there are
(n r + a r - 1) !
(1.2)
n r \(a r -\)\
different ways of partitioning the n r light-quanta among
the cells when we disregard individuality of the quanta.
The formula, in fact, goes back to (2 . 5) as it gives the
number of actual terms in the expansion of
where for the moment we are writing 6 for a r and p for n r .
The coefficient
p\
Pi\p 2 i. . .p b \
of a term such as x^ 1 x 2 p * . . . x b p t> is disregarded. It
entered into the earlier work because we assumed that each
particle had a recognisable individuality. Now we regard
the term x-f 1 x 2 Vt . . . x^ just as representing one way of
partitioning of p light-quanta among the 6 cells, and there
are just (p + b 1) \/[p ! (6 1) ! ] such terms, i.e., ways
of partitioning.
Expressions similar to (1 . 2) give the number of ways of
partitioning the n l light-quanta among the a l cells of th$
APPENDIX ON RECENT DEVELOPMENTS 273
range o to v v the /i 2 light-quanta among the a 2 cells of range
v l to i> 2 > an d so on Any combination of special ways of
partitioning in the separate shells gives one way of partition-
ing the total n light-quanta over the whole phase-diagram.
Hence the total number of ways of distributing n light-
quanta is on this view of the matter equal to the product
of the terms such as (1.2); i.e., it is
n r \ (a, - 1) !
where S n r = n,
and each a r is defined by (I . 1).
There is, of course, the energy condition to be satisfied,
viz.,
Zn r Uv r = ^ ..... (1.4)
where E is a constant ; but there is no constant n condition.
The problem is now to find the values of n v n 2 , . . . n r , . . .
which make (I . 3) a maximum subject to the one condition
(1 . 4). The procedure is the familiar one. Calling (I . 3)
W (n, a), we take its logarithm, use Stirling's theorem,
and apply the variational method.
So we have to solve
8{Z (n r + a r ) log (n r + a r ) n r log n r a r log a r } = 0*
subject to
v r %n r = 0.
Thus using an undetermined multiplier a we easily find
that for each value of the integer r
log (n r + a r ) log n r av r =
and so n r + a r
n r
. .
= exp (av r )
r exp (av r ) 1 (I 5)
Equations (I.I) and (1 . 4) will then determine a as a
function of E and v ; for (I.I) shows that
4 77 V V? 8v r
n r = - - .
c 3 exp (av r ) 1*
* We can drop out minus unity as negligible.
274 STATISTICAL MECHANICS FOR STUDENTS
It follows that
v c 3 exp (av r ) 1
or going to the limit we find for the energy density
exp (av) 1
In this reasoning we have omitted, so far, any reference
to one point which will be familiar to those acquainted with
optical theory, and which has made its appearance in these
pages in Chapter xxn. It is the matter of polarisation
of light. The effect of allowing for this is to double the
above result and make it
exp (av) I
To find a in terms of the temperature 6 of the enclosure
we introduce the usual expression for the entropy, viz.,
k log W m , where
log W m =Z [K, + a r) log (n r + a r ) - n r log n r - a, log a r ]
the n r having the values given by (1.5). If we now raise
the temperature the energy of the radiation increases to
E + SE, and there is a corresponding increase of entropy
from S to S + SS. The limit of 8S/SE is 0" 1 , since the
volume v of the enclosure is maintained constant, and no
external work is done by the pressure of the radiation.
Writing e r for liv r and //, for a/h we have
S = TcZ { (n r + a r ) log (n> + a r ) - n r log n, - a r log
r +a r )(logn r + av r ) n r log n r a r
= kza r (log n r - loga r ) + (n r + a r ) ^
= k 2 [ a r log [exp (p r ) 1] + a, pe r +n r
r (/ie r - log [exp (^ f ) - 1]) j + A/n E
- Za r log [1 - exp (- ^ r )] } .
APPENDIX ON RECENT DEVELOPMENTS 275
So it follows that
= ^
, , , _, du. , ( d log fl exp ( ,)] )
= kp + lcE-- -kS\ a r 8 L j v ^' n
cufj f dfj j
du, j ( a r . . du, ]
^ L ^ ] ' : tXV ( fJL r ) v - f
exp (p, r ) -
and thus JJL has its usual value, viz., (kQ)~ l . So we arrive
ultimately at Planck's expression for the energy-density
in the temperature-enclosure, E/v, viz..
exp (hv/k0) 1-
If. Einstein's Theory of an Ideal Gas. The method of
Bose is therefore justified by its success. Einstein immedi-
ately applied the same method to the discussion of a mona-
tomic gas. The region of the phase-diagram in which will
fall all the points representing any molecule whose energy
does not exceed the value e is given by integrating dx dy
dz d dt] d( throughout a range of values determined by the
volume v of the vessel and the condition
Its magnitude is
3
The region in which lie the points corresponding to mole-
cules whose energies lie between e and e + Se is thus
and the number of elementary phase-cells in this region is
thus
2771; (2m)*
<* *<
276 STATISTICAL MECHANICS FOR STUDENTS
As before, we divide the phase-diagram into shells by the
energy-hypersurfaces corresponding to e v 2 , . . . e r , . . .
and write e r - r _ 1 Se r . The number of cells in the shell
bounded by r _ l and e r is a r where
a^Vse, .... (II. 1).
The partitioning is carried out as above, and the number of
complexions corresponding to an energy content of the gas
equal to E is W (n, a), i.e., expression (1 . 3) where
S n r r = E.
But, of course, we have now the condition to satisfy that
Z n r n constant,
for although we disregard individuality of molecules in
counting complexions, we must remember that in this case
no particles are created or destroyed. When we take the
logarithm of W(n, a), vary it, put the variation equal to
zero and take account of
Z r Sn r =
and E 8n r
we obtain
log (n r + a, r ) log n r = A + /**,,
involving two undetermined constants as usual. Thus
. . (II. 2)
exp (A + ii r ) 1
where, of course, a r is given by (II . 1).
The entropy S is k log W m where W m is the value of
W(n, a) with this expression for n r inserted. It is easily
shown as above that it is given by
S = k [n\ + /xE - Za r log [1 - exp (- A - p, r )] }
(II . 3).
On working out dS/dEt as above, we find, as before, that
the constant ^ is connected with the temperature 6 by the
usual result. Putting in the values for the a r it follows,
APPENDIX ON RECENT DEVELOPMENTS 277
after a step or two, that the number of molecules whose
energies lie in the range e to -f Se is given by
^Sc ... (II. 4)
where the constant A is determined by the condition
2 TTV (2m)'
n v '
The result (II . 4) replaces the Maxwell Boltzmann law,
to which it clearly reduces in case where unity is negligible
in comparison with exp(X + jue). Further it easily follows
that
_ 5)
8 -
%/0
and
Q ^, E 2 TTV (2 m)*k C , . ,_ x weN ,
S = RA + ~ - - ^ > I * log (1 - c-*-*') de
^ (II. 6).
III.' The Fermi-Dirac Statistics. This change in the
manner of counting complexions was suggested, as we have
said, just on the eve of the third period in the history of
quantum ideas. About the same time a suggestion was
made by Pauli in connection with the Bohr theory of atom
structure, which is called Pauli's exclusion principle ; it
concerns the description of the quantum paths of the
electrons in an atom in terms of quantum numbers, and
postulates that in a given atom or molecule no two electrons
can have the same set of quantum numbers, i.e., that at any
instant two electrons cannot be on the same quantum
orbit. In statistical theory we have seen already the
analogy of quantum paths of representative points and
phase-cells, and in 1926 Fermi suggested that in dealing
with the statistics of systems of molecules we should reckon
up the complexions on an analogous exclusion principle,
viz., that complexions in which two or more representative
points occupied one energy cell should be ruled out as
impossible. Independently, Dirac made the same sugges-
278 STATISTICAL MECHANICS FOR STUDENTS
tion, and also showed how it could be regarded as a reason-
able conclusion from the new quantum mechanics, which
was just about a year old at the time, and which had also
produced a theoretical justification of the original Pauli
exclusion principle concerning electrons in atoms. The
effect of this on the formulae of statistical theory is not
hard to discover. Like Bose and Einstein, Fermi and
Dirac abandon any idea of individualising molecules, but
concentrate, as it were, on the individuality of the elementary
phase-cells in an " energy-shell " of the phase-diagram.
For Bose and Einstein the occurrence of any number of
representative points in a cell was as likely as the occurrence
of any other ; for Fermi and Dirac this is not so ; a cell
can contain one point or none ; each alternative is equally
likely ; but it is impossible that it should contain two or
more. The earlier results of Section II. are simply repeated,
viz.,
2 TTV (2m)* i ,> /TTT .. x
a ' = ^r~ L ^ 8c, . . . Ill . i
A 3
where E n r e r = E
and Z n r = n.
But instead of taking W(n, a) as given by (1 . 3), it is
determined by
W(n,o) = n a ' 1 . . (III. 2)
n r I (a r - n r ) \
for the number of ways of arranging n things in a boxes on
the understanding that no box is to hold more than one
thing is just the number of combinations of a things taken
n at a time. The algebraic procedure now follows similar
lines, and produces these results
n r = ?L . . (III. 3)
cp (A +/,) + 1
S - k log W m
(- A -/,)]}
(III . 4)
and, dS/dE being put equal to 0" 1 , we find the usual result
APPENDIX ON RECENT DEVELOPMENTS 279
for jii. The Fermi-Dirac law of distribution replacing the
Maxwell-Boltzmann law now turns out to be
with A determined by
2<rrv(2m)U * ,
n = A_ L ...... dc.
Also
O -_/,, /O /wi\>s / ,5
de . (Ill . 5)
Jo
and
S = RA + ?
(III. 6).
The change from minus unity to plus unity is, of course,
a negligible affair under conditions where the Maxwell-
Boltzmann law is a justifiable approximation, but, as we
shall see in a moment, it is quite otherwise under more
unusual conditions . But even when the Maxwell-Boltzmann
law is a good approximation the new statistical theory pro-
duces the value of the entropy constant in a very direct
and easy manner. Thus if we neglect the ^ 1 we obtain
x r i
n ye M e* e ** ae,
where y is written for 2 TTV (2w)^/& 3 , and so writing # 2 for
fjie and using the familiar results for i # f e"""* 1 dx we find
Jo
n - ye~ T
so that e x = ~ (2 TTW, k9)% . . . (Ill . 7).
nh 3
For hydrogen under normal conditions this gives 6 X 10 4
for A, and so justifies the approximation used, but if v or 6
are very small, i.e., at large densities or low temperatures, A
is too small to permit the neglect of the unity term. Pro-
280 STATISTICAL MECHANICS FOR STUDENTS
ceeding, however, under the suitable conditions we find that
E'=ye~ x r^e-^dc
Jo
1 E
and so
n 2
the familiar energy-partition law.
Hence S = RA + ? + kye~ K f c* eT* dc,
u Jo
remembering that log (1 x) x if x is sufficiently
small compared to unity. Thus
S = RA + + Jen
u
III . 7 = R flog v log n 3 log h +- log
L Q r,-i 2
by
T? T 5 i / i , 5 i i (2 7rm/t) 3 A;~|
- R [-log 6 - log p + - + log-L pJ J
(III. 8)
where p is written for ^A;0 as usual.
To decide between the two results one must consider
conditions under which the unity term is not negligible,
and then experimental evidence leans towards the Fermi-
Dirac result, and so supports the conclusions of the new
quantum mechanics. The trouble with the Einstein result
is that while it makes the entropy of a gas tend to zero as
the temperature decreases to zero (thus extending the
Planck-Nernst heat law to all substances, condensed or
not), it makes the specific heat first rise to a maximum
and then decrease to zero asymptotically. Now this
maximum property has not been observed for any gas, and
APPENDIX ON RECENT DEVELOPMENTS 281
so the Einstein-Bose statistical theory lacks support in that
particular. On the other hand, with the Fermi-Dirac
statistics, it can be shown that at low temperatures
E = E + a 2
where ^ __ 3 /6n\*nh*
~~ 40 (in?) ~m
j o/^" V \* m 9979
and a = 2 ( ) - ?r 2 n 2 k 2
\9n/ rc& 2
so that the specific heat 2 aO approaches zero as a limit
without any intervening maximum. It is clear also that
the Fermi-Dirac statistics necessitates a nul-point energy ;
this is evident from the fundamental postulate, for of all
the molecules whose representative points fall within a
group of phase-cells which have one volume-element in
common only one can be in that phase-cell of the group
which corresponds to zero velocity. All the others will
fall into cells with as small energy as possible, but up
to a certain limit each cell will have one point in it. In
the Einstein-Bose statistics the constant A is essentially
positive, and has zero as its limiting value ; this is obvious
from result (II . 4) ; otherwise n r would be negative for
small enough values of e. At the limit A all the repre-
sentative points would crowd into the cell of lowest energy,
since l/(e^ e 1) is infinite for the zero value of e. This
is an extreme case of what is called " degeneracy." So as A
decreases, either with decreasing temperature or increasing
density, the system passes from the classical Maxwell-
Boltzmann distribution to a state of extreme degeneracy ;
and for small values of A particles with a low velocity are
present in greater numbers than is the case of a classical
distribution with the same density and temperature. In
the Fermi-Dirac statistics, on the other hand, A may have
positive or negative values.* The distribution differs
markedly from the classical when v (2 7rmkd) ? ~/(nh*) is much
less than unity ; A is then large and negative, thus making
* N.B. The values of A are not necessarily the same for the three cases,
even when density and temperature are the same.
282 STATISTICAL MECHANICS FOR STUDENTS
small compared with unity, except for very large
values of e. This means that for a large range of e, n r = a r
(see III . 3), or every cell has one representative point in it.
The system is degenerate. In cases of degeneracy the
Fermi-Dirac statistics implies that molecules with low
velocities occur in greater number than in a classical dis-
tribution at the same temperature and density.
The reader should for further information consult a
paper by Lennard-Jones in the Proc. Phys. Soc. (London),
Vol. XL., Part 5, (August, 1928), where a full bibliography
of papers on these matters will also be found.
IX, The Statistical Method of Darwin and Fowler. Parti-
tion Functions. An innovation of a different character has
been introduced lately by Darwin and Fowler. The
original papers are to be found in the Phil. Mag., Vol.
XLIV., Nos. 261, 263 (September, November, 1922).
They begin by pointing out the weak spot in the practical
working out of the statistical calculations on the usual
lines, viz., the use of Stirling's Theorem. This point is not
always brought home in current accounts. Let us, for
example, consider a gram-molecule of a gas under standard
conditions with a volume about 20 litres. The energy is
1*5 R0, about 3 x 1C 10 ergs. If this were all in one molecule
its momentum would be (1-5 Rra0)*, about 10~ 7 grammem,
for hydrogen. Thus the whole momentum extension
would be of the order 1C"" 20 in the phase-diagram, and when
associated with the volume 20 litres, the phase-extension
of the phase-diagram actually required for a gram-molecule
of hydrogen at ordinary temperatures and pressures would
be of the order 10"~ 16 (erg-sec.) 3 . Suppose this is now divided
into elementary cells, h* each having, therefore, an extension
of the order 10~ 79 . There are 10 83 such cells in the whole
extension. But there are only about 10 24 representative
points to partition among them ! But the use of Stirling's
theorem quietly assumes that in practice the points avail-
able are much more numerous than the cells.
Of course, on purely classical lines we can retort that we
are not compelled, in order to arrive at the classical result,
to postulate cells as small as Planck's element. Still the
APPENDIX ON RECENT DEVELOPMENTS 283
use of that element in quantum discussions of the " classical-
quantum " type requires subsequent justification of the
results.
Darwin and Fowler show that this rather illegitimate use
of Stirling's theorem can be avoided entirely by considering
the average state of a system, rather than its " most prob-
able/' as the analogue of the state of thermodynamic
equilibrium. In practice this leads to no new result, but
the point of view and method are quite different. We can
illustrate this by a simple example. Suppose we have
present a set of Planck vibrators which each contain a
multiple of a unit of energy . Let there be n vibrators
and an amount of energy E available where E/e is an
integer. Let us consider a complexion in which there are
n vibrators with no energy, % with , n% with 2e, etc.
There are as usual
n 1
n 1 n^ \n 2 \ . . .
complexions in a corresponding statistical state, and we
have to satisfy the conditions
n + n l + n 2 + n 3 + = n Ty
1 + 2tt a + 3n 8 + . . . =E/ l ' ''
Let C be the total number of complexions compatible
with these conditions. C will be given by
nl
C = : : i
n ! % ! n 2 ! . . .
where the summation is over all possible values of the n
which satisfy (IV .1). If we consider the series
(1+2 + z 26 + z 36 + . . .) n . . (IV . 2)
the typical term is
where the n r satisfy (IV .1). Thus if we pick out the
coefficient of 2 E in the expansion (IV . 2) we shall have
the value of C. So C is the coefficient of Z E in (1 z )"~ n ,
284 STATISTICAL MECHANICS FOR STUDENTS
i.e., of y in (1 y)~ n where c = E/e. But by the binomial
theorem
1. Z. 1. Z. o.
(^_+c_-^!
^ (w - 1) ! c ! y ^
and so p __ (n + c 1) !
~~ (n- l)!c!
a result which we have already met several times.
The average value of any quantity denoted by u can
then be determined by the equation
Cu =2 _ _ uz (n l + 2n t + zn, + ...)< m (IV . 3).
n I n 1 ! n 2 I . . .
Thus the average value of the energy is given by
,
n \ n \
<fe v '
d I
z
dz (1 z') n
2 ~ log (1 - z )l (IV . 4).
dz J
The expression (IV . 4) is obtained so as to render possible
a method of approximate calculation which discards Stirling's
theorem and uses the method of " contour integration." *
In this z is regarded as a complex variable, and by integrat-
ing round a circle in the Argand diagram with the origin as
centre, and a radius less than unity, we know that
* See Jefferies' Operational Methods in Physics (Camb. Math. Tracts).
APPENDIX ON RECENT DEVELOPMENTS 285
and
These expressions are exact, but in the theory of contour-
integrals there is a method known as the " method of
steepest descents " which leads to simple approximations
that are adequate and rigorous under certain conditions
satisfied in most problems. The process is this :
The integrands being infinite at z = 0, and also at z = 1,
there must be a minimum value of the integrand at some
point on the positive axis where z = ft (ft is of course, a
real number less than unity). Now take the circular
contour round which the integration is effected to be a circle
of radius ft. It can be shown that as we travel along this
circle the values of the integrands have strong maxima
at z ft and drop to relatively very small values at all
points on the contour outside the very short arc containing
the point z = ft. Lest there be any confusion on the part
of the reader about these statements, let him note that the
value of an integrand at z = ft is a minimum for all the
values at points on the real axis between z = and z 1.
But it is maximum for the values at points along the circle.
Indeed, if n and E are large enough this maximum is so
strong that practically the whole of the contour integral is
contributed by a short arc of the circle in the neighbourhood
of z = ft. Under these circumstances we can remove the
term n z(d/dz) log (1 z e ) outside the integral sign provided
z is given the value ft.
This value ft is thus obtained by solving the equation
i.__j =
7 Tt 1 4- I / i *\v) '
or practically
d l = 0.
_ _ _
dz z E (l
This equation is easily seen to be
E (1 z') UZ* = 0,
286 STATISTICAL MECHANICS FOR STUDENTS
so that & is given by
We then have, owing to these approximations
c = A
dif
where A is some constant.
So that
Now, of course, this result is entirely trivial in this case.
As we assumed that the energy of the system is constant,
obviously the average energy of the system over all the
complexions is just the constant energy for each complexion,
as (IV . 6) and (IV . 7) show. The example, however, is
the easiest illustration of the process of approximation
employed. Let us choose one which does not lead to a
trivial result, viz., a system which consists of a vibrators
with a fundamental quantum energy e, and 6 vibrators with
a fundamental quantum 77. For the purposes of the calcula-
tion we must assume that e and 77 are commensurable, but
it does not matter how large are the integers which are
required to express 6/77 in its lowest terms. We may, of
course, choose the unit of energy so that e and 77 are them-
selves integers. The number of complexions embraced in
a state in which a r vibrators of the first type have an
energy re and b g vibrators of the second type have an
energy 577 is
APPENDIX ON RECENT DEVELOPMENTS 287
where we must satisfy the conditions
Ea r = a
Sb 8 =b
S (ra r + sb 9 r)) = E . . . . (IV . 9).
The total number of complexions possible with the energy
E is as before
C=27 _ ^ __ _ ^ _ (IV. 10)
ajaja,!. . . 6 !6 X !6 2 !. . . V '
where the summation is over all values of the a r and 6,
compatible with (IV . 9). Now the whole energy E can be
considered as made up of two parts, E a and E 6 , one the
energy in all the vibrators of the first type, the other in
those of the second type. The average value of E a over all
the complexions is given by
- K + 2a 2 + 30 3 + . . .) . a ! b I
-
2 E n
As before, we can easily show that C is the coefficient of
1
(1 - Z e ) (1 - Z") 6
and C E is the coefficient of S E in
1 d 1
z
(1 z' 1 )" dz (I z') a
d_
or 1 dz
(1 z"f (1 z') a
This leads to the following contour integrals for C and
CE fl
= 1 t_dz 1
~'2m}~it & + 1 '(l*Y(l -
dz fflz ^
2m Z E + 1 (1 - z') a (1 -
288 STATISTICAL MECHANICS FOR STUDENTS
There is a strong maximum for the integrands in a small
arc at z = ft of the contour circle of radius ft where ft is
determined by
d 1 ; = o
(IV. 12).
v '
dz 2 E + 1 (1 z e ) a (1
so that & is found from the equation
This leads to the approximation
A
C
ft e ) a (1
d
^-a- a E+1 (l-ftT(l-;
leading to
9--'- 1
similarly,
~V 1/1 Q.cN
(IV . 13)
This determines the average partition of the energy
between the two types of oscillators. The method can
obviously be extended to any number of different types.
We have to identify the quantity & with the usual properties
of the system. It is fairly evident that it is connected with
the temperature, and if we take the temperature on the
absolute thermodynamic scale to be the same as that
defined by the equipartition law for systems following the
classical dynamical principles, we can find what 0- is. Thus
suppose that the second type of vibrators have a low
frequency and a small value of 77. In the limit they follow
the classical laws, and ^"^ is nearly unity. If it is a little
greater than unity
log [i + (#-' - i)] == a-* _ i
APPENDIX ON RECENT DEVELOPMENTS 289
or $- 1 = TJ log S-,
so that t\ _._ 1
"
-'-! " log
If ft" 11 is a little less than unity
log [1 - (1 - -')] = -(1 - *-'),
and the same result follows. Thus in the limit the particles
of a system obeying classical laws have each an average
energy, log (!/&) This is, of course, kinetic and potential,
and is &0, so that
=e- fj -
where ,u is (&0)" 1 , and we arrive at the usual Planck law of
partition among the oscillators of the first type, viz.,
E - a
a
A generalisation to more general types can be easily
made. Suppose that particles of type 1, a in number, can
be in states for which the energies are e 1? e 2 , e 3 , etc. ; particles
of type 2, b in number, can be in states for which the energies
are rj ly r) 2 , ^ 3 , . . . and so on. Further, we may suppose
that several quantum states belong to one energy state, so
that the states of type 1 particles have weights p l9 p 2 ,
p 3 , . . . respectively, of type 2 particles, q l9 q 2 , g 3 , . . .
respectively, and so on. The weighted number of com-
plexions which correspond to an assigned specification is
a\ ai fli 6! bi ,
where
The functions
j(z) =p 1 z\+p 2 z e * +Pz#* + -
+ (z) = qi z* + ?a z" + q 9 z*> + .
take the place of (1 ?.*)~ 1 9 (1 2 1 )"" 1 , etc., in the previous
290 STATISTICAL MECHANICS FOR STUDENTS
example. They are called the " partition functions " of the
corresponding types of particles. Proceeding as before, we
find that C is the coefficient of Z B in
( Z )] [V (2)] . . .
and C E. is the coefficient of Z E in
^ft <*)]} [*<*)]*
i.e., in
/ j \
ft(*
These are converted into contour integrals, and we have
ft ()]'[()]*...
We determine the radius & of the special circle of integration
by finding the condition for the minimum of the integrand
in C, i.e., by solving
dz
i.e., & practically satisfies
E = a$ -- log <(&)+&& A log f (&) + ... (IV. 14).
On putting this value of & into the contour integrals
above, we again find that approximately
and (IV . 15)
<a ft log </> (#) | ft ($)]"
C E fl = A
APPENDIX ON RECENT DEVELOPMENTS 291
so that ^; n d , i /ft \
a = a g * ( J
(IV . 16)
etc.
In applying this method to a system containing free
molecules where quantum states do not strictly enter, we
can consider the various states defined as usual by elementary
cells of the phase-diagram 8xSySzSSr)8 * for which the
energy is ( 2 + r) 2 + C 2 )/2w. The partition function is
then
where 8a is written shortly for the sextuple differential
&g...S, and the suffixes correspond to the various cells ;
e, is ( r 2 + rj r 2 + r 2 )/2w, and the weight of each state is
defined to be 8a/h 3 which we have already seen to be the
justifiable method for combining non-quantum parts with
quantum parts of a system, on the understanding that
each quantum state is given the weight unity. In the
limit x (z) is a sextuple integral, and becomes
where v is the volume of the vessel.
Thus suppose there are present a vibrators with partition
function </> (z) and n free molecules. The parameter 0- is, as
before, determined by
E - ab ~ log < () + n -| log x (),
Ci \T U> t7
and the energy in the n gas molecules is, on the average,
equal to
* N.B. Do not confuse the z co-ordinate of a molecule here with the
z used as the symbol for the complex variable.
292 STATISTICAL MECHANICS FOR STUDENTS
Now 2 <:=:e< logz
_ log z _ log (1/z)
where a =
2m 2m
400
SO X () = -^
_ (2 77-ra) 3 ;
~ * [log (l/z)p-
Hence the energy of the gaseous part of the system is
equal to
d ! (2 Trmf* v
log
_ 'A 8 [log (I/ft)]*
ft d i 3 lo
dftJ2
371 1
2 log
But by the usual elementary result of kinetic theory the
pressure of the gas is two-thirds of its energy per unit
volume, and so
_n I
P ~v log (I/ft)'
and since by the usual definition of temperature
n -, *
p=-kO,
v
we thus arrive once more at the relation between ft and
temperature
log (I/ft) -1 =VL
or ft = e-.
We can, in a similar manner, work out other mean values.
Thus the mean over all complexions of the number of
APPENDIX ON RECENT DEVELOPMENTS 293
molecules or vibrators of the first type in the r th quantum
state is obtained thus
C a, = Za r , "' Pi 1 P Z "' y-n^ - i 6 ' if* .
' r aj ! a z l . . . b\ b z \ . . .
= ap r E , ( . a ~ 1 . )! - . Pfpj* 2V'- 1
*" a>i\ a z \ . . . (a, 1) !
where the summation in the second line satisfies the energy
condition
a l l + a 2 + . . . + (a r l)<E r + . . . 4 2b t rj t = E e r
It follows that
-q^ A
f),JS - e,- I- 1
From the value of C in (IV . 15) we find that
Thus a r is proportional to & er or, e""^ r , and we have the
usual Planck partition law, degenerating to the Maxwell law
for gas molecules following the classical laws.
Another important mean value is the mean of the
squared energy, E a 2 . Reviewing the argument by which we
obtained E a , the reader will see that we shall obtain E a 2 if
we employ the operator (z ) instead of z . Thus
_ v dz 1 dz
C E a 2 is the coefficient of 2 E in the expansion of
and therefore, is equal to
jfrci]'---,
~
i (
2 mJ
294 STATISTICAL MECHANICS FOR STUDENTS
which is approximately equal to
[* (*)]*
Hence, using the value of C in (IV . 15) we have
o*
ftw
I
- . . . by (IV . 16)
The fluctuation of the energy is measured by the mean
value of the square of the difference E a E a . Now
(E,- EJ = E a 2- 2 E. Ea + (E,) 2
and the mean value of this is equal to
APPENDIX ON RECENT DEVELOPMENTS 295
dp,
dd
de '
a result which Einstein made use of in treating fluctuations
of radiation in a temperature enclosure in the early days of
quantum theory. The legitimate use of the approxima-
tion in the case of the operator (z d/dz) 2 requires careful
analysis, but it appears that it is quite rigorous in a tempera-
ture bath.
In the Proc. Eoy. Soc., A 113, p. 432 (1926), will be
found a paper by Fowler reviewing recent statistical theory,
and showing how the method of partition functions can be
applied to the statistics of Einstein and Bose, and that of
Fermi and Dirac.
APPENDIX ON COLLISION-FORMULA AND
CHEMICAL KINETICS
IN the treatment of systems in statistical equilibrium it is
postulated that energy can be transferred from molecule to
molecule, but no assumptions concerning the mechanism by
which energy is so transferred are required. When, how-
ever, we deal with systems which are not in equilibrium, it is
only natural to expect that we shall have to take more
careful account of the nature of the forces which act between
molecules ; for while the ultimate state of equilibrium
attained is independent of the special laws of the intermole-
cular forces, the manner in which that state is approached
and the rate of transformation is clearly dependent on them.
The problems raised by such considerations hardly come
within the scope of a book which is an " Introduction," and
which has already grown to the limits set by the capacities
and needs of its probable readers. However, it may be of
some value if a further appendix, which will not seriously
encroach on the student's time and patience, is added, and
one or two matters which are fundamental in the treatment
of non-equilibrium states are dealt with. They concern the
frequency of molecular encounters and the bearing which
this has on chemical kinetics.
I. Collisions between Molecules in a Gas. In the Kinetic
Theory of Gases a very simple type of intermolecular action
is assumed for many purposes. An encounter (that is the
interval during which two molecules are within the sphere
of one another's action) is considered to be so brief in
relation to the time of free path that it is pictured as a
" collision " which takes place instantaneously when the
centres of the two molecules are separated by a definite
distance which we denote by a. In short, the molecules
are visualised as " hard spheres," a being equal to the
296
APPENDIX ON COLLISION-FORMULA, ETC. 297
diameter if they are like molecules and to the sum of their
radii if they are unlike.
For the reader's guidance through this section, which
involves some rather tedious steps, it may be as well at the
outset to give him a preliminary summary to the various
formulae obtained in it.
(I. 1) is a general result for the number of collisions per
unit time between two types of molecules in a gas mixture
whose velocities are confined to narrow ranges of velocity.
(I. 2) is the specialised form of (I. 1) when the mixture is
in statistical equilibrium.
(I. 3) is the total number of collisions per unit time in the
mixture when in equilibrium.
(I. 4) and (I. 5) are formulae for the total number of
collisions in a gas consisting of one type of molecule.
(I. 6) is a value for the mean free path of a molecule in a
gas.
(I. 7) and (I. 8) refer to the collisions which involve
a relative velocity between colliding molecules which lies
within narrow limits or is greater than an assigned value.
(I. 9) and (I. 10) are similar formulae which involve the
relative velocity of approach, i.e., the component of relative
velocity along the line of centres of the colliding molecules.
The last four formulae are of considerable value in the
problem of chemical reaction in gases.
In treating the problem of collision-frequency we consider
an enclosure in which there are present N^_ molecules of one
type per unit volume and N% of a second type also per unit
volume. We introduce a velocity diagram partitioned as
usual into cells ; at a given instant let the number of
representative points of molecules the first type in the rth
cell be v lr Sa> r , where 8aj r is written for 8u r 8v r 8w r (u r , v r) w r
are the components of the velocity corresponding to the
central point of the cell). The number of the second type
is v 2r 8aj r . We are not considering at the moment an
equilibrium distribution, but there is some law of distribu-
tion, and so v lr will be some function of u r , v r , w r . Represent
it by fi (u r , v r , w r , t) ; the time-variable must, of course, be
involved if the distribution is non-equilibrium. Similarly,
298 STATISTICAL MECHANICS FOR STUDENTS
v 2r is equal to/ 2 (u r , v r , w r , t), / x and / 2 not being of necessity
the same functional form. If the distribution should be ofte
of statistical equilibrium the function/! (u y v> w, t) would be
the Maxwell function
- i pm l (u* + v* + t
and / 2 (u, v, w, t) would be
N 2 (^\* exp r / , m2 ( U 2 + V 2 +
(See expression (4.1.1) and equation (4.1.8), remembering
that = mu, etc.) Our first problem is to obtain the
probable number of collisions per unit time between the
molecules of the first type and those of the second.
First of all let us analyse the relative situation of two
molecules, with their representative points in the ath and
in the 6th velocity-cells respectively, which will lead to a
collision within an interval 82. Let the reader be careful to
guard against confusing situations of representative points
in velocity-cells with situations of the points by which we
sometimes idealise the molecules themselves in the enclosure.
The two cells might be very far apart indeed in the velocity-
diagram without necessarily implying the impossibility of a
collision within a very brief time of many of the molecules
represented. It will be realised after little thought that one
important factor in settling the possibility of a collision
within an interval St between two assigned molecules is the
relative velocity u b u a , v b v a , w b w a i another is the
angle between the direction of this relative velocity and the
direction of the line of centres of the two molecules from the
centre of the b molecule to the centre of the a molecule. If
this angle is denoted by 0, and if or represents the sum of the
radii of the two molecules, there will be a collision within the
interval 8t if the centres of the molecules are now separated
by a distance which lies between a and a + f cos 08t, where r
is the magnitude of the relative velocity and is equal to
necessarily less than a right angle.) Thus the centre of any
molecule of the group b which would collide within time 8
APPENDIX ON COLLISION-FORMULA, ETC. 299
with a specified molecule of group a, the " line of centres
angle " being between and 0+86, would have to lie in a
ring-shaped space whose circumference is 2 ira sin 0, and
whose section is an elementary rectangle with sides a80 and
r cos 08t. The volume of this ring is 2 TrcrV sin cos 80 8t.
Let us assume that 32 is chosen to have such a value that
this ring contains on the average one molecule of the group 6,
so that it is therefore equal to (v 2b 8co b ) ~ l in volume ; then
8t must be equal to
1
v 2b 8aj b 2 770- V sin cos 80
and in this time there is on the average one collision between
a specified molecule of the group a and any molecule of the
group 6 having the defined line-of-centres relation. So in
one second there are
v 2b 8oj b 2 7ra 2 r sin cos 80
collisions between a specified molecule of the group a and
any molecule of the group 6 having the defined line-of-centres
relation. Integrating with respect to between the limits
= o and = 7T/2, we obtain the number of collisions per
second between a specified molecule of the group a and any
molecule of the group b. The result is
V 2b 8a) b 77CT 2 r.
As there are v la 8a) a of the first type molecules in the ath
phase-cell, the number of collisions per unit volume per unit
time between molecules of the first type in the velocity-
condition represented by the ath velocity cell and molecules
of the second type in the velocity-condition represented by
the 6th phase-cell is
77(72 v la V 2b r & w a 8o>&
that is,
2 /i (u a , v a , w a )f 2 (u b , v b , w b ) r ^ (I j.
du a dv a dw a du b dv b dw b * ' '
The total number of collisions per unit volume per second
between all the molecules of type 1 and all those of type 2
will be given by the integral of the expression (I, 1) between
the limits plus and minus infinity for the six velocity com-
300 STATISTICAL MECHANICS FOR STUDENTS
ponents. Of course, such an integration could in general
be carried out only by rather laborious methods of approxi-
mation. When the gas mixture is in equilibrium and we
can use the Maxwell law of distribution, the integration can
be effected by means of the table of intergrals on p. 17,
though not quite so directly or simply as might appear at the
first glance. The trouble arises owing to the appearance of
the factor r, i.e., { (u b u a ) 2 + . . + . . |Mn (I. 1), but the
difficulty can be surmounted by a transformation of variables
and, in order to explain this point, we shall have to digress
for a moment and introduce the reader to two necessary
lemmas.
The first is very simple. Let a, j8, y represent the com-
ponents of the velocity of the centre of mass of two particles
m I and ra 2 , and let , T?, stand for the components of the
relative velocity r, so that
a =
etc.,
and = u b u a
etc.
It is then easy to show that the combined kinetic energy of
the two particles which is
i m i ( u a + v <r + w <?) + i 2 ( V + V + *V)
is equal to
J (m, + ro a ) (a' + p* + y 2 ) + | m 12 (? + ^+ 2 )
= \ ( m i + m*) c * + i m i2 r *
where c is the velocity of the centre of mass and m 12 =
m x m 2 l(m l -f- ^2)- This result enables us to write
fi(u a ,v a ,w a )f z (u b ,v b ,w b )r
which appears as part of (I. 1) in the form
when we are considering the equilibrium distribution.
The second theorem is concerned with substituting for
APPENDIX ON COLLISION-FORMULAE, ETC. 301
the sextuple differential du a .... dw h in (I. 1) an expression
involving the sextuple differential dad^dyd^dr^d^ Con-
sider for the moment a plane diagram and a point A on
it which moves over a certain extension in the plane. Let
x, y stand for the current co-ordinates of A, and let us
consider another point B whose co-ordinates (X, Y) are
connected with those of A by the relations
X = kx + ly
Y == KX + Xy
where k, I, K, A, are any constants. The point B will move
over a second extension in the plane as A moves over the
first. It can be proved that the area of the second extension
bears to that of the first the ratio kX K\. This is most
readily seen by noting that if A 7 and A" are two positions of
A, and B' and B" are the corresponding positions of B, then
the area of the triangle OB'B" is equal to
i (X' Y" - X" Y')
= i {(**' + ly 9 ) (KX" + Xy") - (Tex" + ly") (KX' + Xy')\
= i (iA - K l) (x'y" - x"y')
= (kX Kl) x area of the triangle OA'A".
The result stated above follows when one recalls that any
enclosed area can be divided into elementary triangles having
a common vertex within the area.
If we now consider a two-dimensional diagram in which
we represent u a and u b by a point A, and a and by a point B,
it follows that an extension in the diagram embracing a
continuum of values of u a and u b and the extension em-
bracing the corresponding values of a and are equal in
area, since in this case
As this applies to elementary extensions just as much as to
finite, we can replace du a du b by d&dt; in an integral. A
similar result holds for the other components.
After this digression we can return to the general expres-
302 STATISTICAL MECHANICS FOR STUDENTS
sion (I. 1), and, in the case of the state of equilibrium, write
it as
- | M { (m, + m 2 ) c* + m l2 r*}] r
da. d^ dy dg drj d . . . (I. 2)
The integral of this over all possible values of a, /?, y, |, 77, ,
can be separated into two triple integrals, which, apart from
the initial factors, are
I ex P [""
and
III 6 ^ [~~
By a familiar transformation the first becomes
f
4 77 e#p [ | /x (m l + m a ) c 2 ]
Jo
the second
-00
4 TT I eo:^> [ - J fji m 12 r 2 j r 3 dr
'o
The first of these has the value
W V 2 V
77 . ^1 1
4 \fji (mi + m 2 )/
(See the table of integrals on p. 17, No. 2.)
The second has the value
2 Vju, m lz /
(See the table, No. 3.)
Thus the expression (I. 2), when integrated, becomes
which simplifies to
. . . (1.3)
where we recall that \L = (&#) " a , m 12 = m^ m^Km^^ + ^2)?
and or is the sum of the radii of the two molecules. This
expression (I. 3) is the total number of collisions per unit
volume per unit time between molecules of the first kind
APPENDIX ON COLLISION-FORMULA, ETC. 303
and molecules of the second kind in a gas mixture which is
in a state of equilibrium.
We can easily obtain from this the number of collisions
in a simple gas whose concentration is N molecules per unit
volume. The formula (I. 3) will give it if we put N l equal
to N 2 , m l equal to m a , and divide by 2 ; for, as it stands,
(I. 3) would count each collision twice. Thus the collision-
frequency in unit volume of the gas is
..... (1.4)
2 A72/ 473r ^M
or a 2 N 2 ( - ) 2
\ m J
where M is the gram-molecular mass, and R the gram-
molecular gas constant.
This can be thrown into another very useful form. At
the end of Chapter IV., we calculated the mean value of
squared velocities of the molecules in a gas in the equilibrium
state. We can just as easily calculate the mean velocity.
If we now use the symbol c to represent the velocity of a
molecule the number per unit volume whose velocities lie
between c and c + Sc is
4 77 N ( )^ exp ( /xrac 2 ) c 2 Sc.
Hence the average velocity is
47rf V 2 \exp ( | /Ltmo 2 ) c 3 Sc
which, by using the third integral in the table on p. 17, is
equal to
304 STATISTICAL MECHANICS FOR STUDENTS
Hence the expression (I. 4) for the collision-frequency in
unit volume is equal to
where c is the average molecular velocity.
Since each collision terminates two free paths there are
-y/2 7ra*N 2 c free paths described in unit time by the N mole-
cules. This gives for the average interval of time between
collisions the value
1
V2 77
and if we multiply this by c we have an estimate for the
average length of a free path ; it is
..... (L8)
There are other methods of calculating mean free path,
but they all give values approximately to
7
Returning to the expression (I. 2) and only integrating
with respect to a, /?, y, we can obtain the number of collisions
between molecules of type 1 and molecules of type 2 in a
mixture whose relative velocity lies between limits r and
r -f Sr ; it is
A **
4 TT
~
V 477 2 4
X 4
that is
a 2 ^ ^ 2 (2 TTjLAiO* ea:p [- | />tm 12 r 2 ] r 3 Sr . . (I. 7)
a formula of considerable value in connection with the
problem of chemical reaction in gases.
If (I. 7) is integrated for all values of r from a definite
value from r to oo , we find the number of collisions in which
the relative velocity is greater than r . Now
APPENDIX ON COLLISION-FORMULAE, ETC. 305
f e-""r*dr
Jr o
i r
= -| e-^ydy
^ J y
(where y is written for r 2 )
1 ( f* f * \
= ~ e"(fy- d(e-"*y)[
2a (J y^ Jy o }
(integrating by parts)
a
Hence the number of collisions in which the relative velocity
is greater than r a works out to be
A
C
which by (I. 3) is equal to
. (1.8)
where C stands for the total number of collisions per unit
time.
It is also of interest in connection with chemical kinetics
to obtain the number of collisions in which the relative
velocity of approach of the colliding molecules, i.e., the
component of r parallel to the line of centres, is within a
narrow range of values. To do so we revert to the considera-
tions leading to (I. 1) and (I. 2) and observe, that had we not
integrated with respect to 6 at an early stage, we would
have found that the number of collisions between molecules
with relative velocities between r and r + 8r and line-of-
centres angle between 9 and + 89 would be given by
multiplying (I. 7) by 2 sin 9 cos 9 89, i.e., it would be
2 o- 2 N l N 2 (2 77 ^ 3 m 12 3 )* exp [ J p. m 12 r 2 ] r 3 sin 9 cos 98r89
306 STATISTICAL MECHANICS FOR STUDENTS
In such collisions the component of relative velocity parallel
to the line of centres is practically r cos ; denote it by s.
If this has to be greater than s , then r must certainly be
greater than s , and, for a given value of r, cos 6 must lie
between unity and sjr. So we must integrate the expression
just written first with respect to 6 between the limits 6 = o
and = cos ~ * (sjr), and then with respect to r from s to
infinity. By methods similar to that used for obtaining
(I. 8) we find the result to be
C exp [- fji m 12 s*] .... (I. 9)
If the component of relative velocity is to lie within a
narrow range of values s to s + 8s, the number is given as a
differential of (I. 9) ; it is
2 fjnn lz C exp [ /*w 12 s 2 ] s Ss . . . (I. 10)
II. Collision-Frequency and Equilibrium. The H Theorem.
The formula (I. 1) is quite general, and in deriving (I. 2)
we assumed the Maxwell form for the function / (u, v, w).
As a matter of fact, the Maxwell distribution law for a gas
in equilibrium can be deduced from (I. 1), and, indeed, such
methods of deduction were the first to be employed before
the work of Boltzmann, Gibbs and Jeans had revealed their
inherent weakness, and in doing so developed the methods
of Statistical Mechanics which are " rigorous " in the true
meaning of that word ; viz., deduced with a clear perception
of the assumptions which we are making in the proof. A
few words about this matter will not be out of place.
Considering the expression (I. 1) for a gas with one type
of molecule, viz.,
*f ( U a> V a> W a) f K> v b> w b) r d "a da >b>
we see that if it were integrated for all values of u b , v b , w b >
between plus and minus infinity, the result would be the
rate at which molecules whose representative points are in
the ath velocity-cell are leaving that velocity-state owing
to encounters. It is theoretically possible, if we know
enough about the molecules, to determine also the rate at
which molecules are entering that velocity-state from other
states. For that purpose it is essential to know sufficient
APPENDIX ON COLLISION-FORMULA, ETC. 307
facts about the law of the forces between the molecules.
As before, the simplest assumption is that of an instantaneous
collision, combined also with the further assumption of
perfect elasticity which implies that no energy of translation
is converted into internal energy, i.e., that the spheres are
" hard and structureless."
There is no space to go into details of the proof (the
reader will find them in full in Jeans' Dynamical Theory
of Oases, Chapter II.). They show how one can find by
dynamical methods the equations which give u c , v c , w c , u d ,
v d , w d and ^ in terms of u a , v a , w a , u b , v b , w b and 0, such that a
collision between a molecule in the cth velocity-cell and one
in the dth cell with a line-of-centres angle between <f> and
(f> + 8(f> will result in one molecule entering the velocity-
state a and the other the velocity-state b, the line-of-centres
angle lying between 9 and 6 + 89 (with the molecules, of
course, separating and not approaching). The number of
such collisions, is, of course,
277 a 2 f (u c , v c , w c )f(u d , v d , w d ) r' sin c/> cos <S0 8a) c Saj d
where r' is the relative velocity of molecules of the class c to
those of the class d. Concerning this expression there are
two important remarks. First of all, owing to the equations
referred to above, / (u c , v c , w c ) f (u d , v d , w d ) can be expressed
as a function of u a , v u , w a , u b , v b , w b and 0. Secondly,
by an application of the Liouville Theorem, it can be shown
that the diffcrentical expression r' sin </> cos </> 8(f> Sco c So> (/ can
be replaced by r sin 9 cos 9 89 8w a 8a) b . It therefore follows
that the net rate at which molecules in the velocity-state a
are gaining in number at the expense of other states is
obtained by integrating the following expression for all
values of u b , v b , w b , between plus and minus infinity and for
values of 9 from o to 77/2
{ / (u c , v c , w c ) f (u d , v d , w d ) - / (u a , v a , w a ) f (u b , v b , w b ) }
2 Tier 2 8co a r sin 9 cos 9 dw b d9 .... (II. 1)
Now if the state is one of equilibrium this integral must
be zero, and one very obvious method of effecting that is
to make the expression within the { } brackets equal to
x 2
308 STATISTICAL MECHANICS FOR STUDENTS
zero. Proceeding on that line for the moment, it follows
that
lo g/ (^a> V a W a) + ^g/ (u b , V b , W b )
= log/ (u c , v e , w c ) + log/ (u d , v d , w d ).
It then appears that the only way to satisfy this functional
equation consistent with the dynamical relations which
connect u a , . . . , w b with u c , . . . , w d , is to write the function
/ (u, v, w) equal to
A exp [ p{(u u )* + (v ~ v Y + (w w o y\,
where A, JJL, u , v , w are constants. This is essentially a
Maxwell distribution for velocities u u o9 v v 09 w W ,
which simply means that it refers to a gas in statistical
equilibrium whose centre of gravity is moving with a
uniform velocity u , v , w , in our frame of reference.
Now from our earlier investigations we know that we
were bound to reach this conclusion for equilibrium with any
mechanism which obeys dynamical laws, let alone the very
simple one we have postulated. Nevertheless, without this
general support for the argument, one must admit one
weakness in it. We have, for equilibrium, to equate the
integral of (II. 1) to zero, and we adopted one very obvious
way of doing this, but it is by no means obvious that it is the
only way. It is on general grounds quite possible that
functional forms of f (u, v, w) might exist which without
making the integrand zero for all corresponding values of
the variables would, nevertheless, make the integral of it
over the whole range of those variables vanish. In that case
equilibrium would be preserved without the " detailed
balancing/' as it is called, which would prevail if we make
the more simple assumption referred to. A great deal of
discussion on this matter took place during the last years of
the nineteenth century. At length Boltzmann propounded
a famous theorem, called the H theorem, which ostensibly
settled the matter by showing that for elastic collisions, at
all events, detailed balancing was the only method of
preserving the gas in its equilibrium state. Yet it was
pointed out in the discussion that Boltzmann's Theorem
apparently was in flat contradiction with the reversibility
APPENDIX ON COLLISION-FORMULAE, ETC. 309
inherent in dynamical occurrences. In the clearing up
of this point and in the strict determination of the sense
in which Boltzmann's Theorem is true, one may say
that Statistical Mechanics was born and methods for
the treatment of molecular phenomena without resort
to any dynamical details of intermolecular collisions de-
veloped.
Boltzmann, in his proof of the H theorem began by con-
structing the function
f(u, v, w) log/(^, v, w) dudvdw . . . (II. 2)
the integral extending over all values of the velocity
components between plus and minus infinity. This is really
a function of t ; for the reader will remember that in cases
of non-equilibrium the time-variable should really be
included among the variables. Boltzmann denoted it by
the symbol H. If we differentiate the expression with
regard to the time, we find that
The value of df/dt for a particular value of u, v } w, can be
derived from (II. 1) in the manner indicated above, elastic
collisions being postulated. The details can once more be
found in Jeans' book. The upshot is to prove that dH/dt
can only be zero or negative, and that if it is zero, then
necessarily
f K> v a , w a ) f (u b , v b , w b ) = f (u c9 v c , w c ) f (u d) v d , w d ) (II. 2)
Now H depends solely on the law of distribution of the
velocities at the moment and so remains unchanged if the
law of distribution remains unchanged. Thus, if there is
equilibrium it follows that dH/dt must be zero, and it equally
follows that (II. 2) must be true, and detailed balancing with
Maxwell's law is deduced as necessary for equilibrium.
This is satisfactory so far as it goes, but the further part of
the theorem; viz., that if dH/dt is not zero, it must be
negative, which appears to be logically bound up with the
other part, gives us pause. If it is true, then the most
310 STATISTICAL MECHANICS FOR STUDENTS
obvious interpretation is that if the state is not an equili-
brium one, the function H will continually decrease until it
reaches a minimum value and retain this value thereafter,
the system having in the meantime attained equilibrium.
Now this is inconsistent with the dynamical principles from
which it has been presumably deduced, for by these principles
any motion is reversible ; so if a certain state of motion is
possible for the system, that state is also possible in which
the position of each molecule at any moment is unchanged
and its velocity exactly reversed in direction. But after
such reversal, the system, assumed to consist of perfectly
elastic spherical molecules, would exactly retrace the " path "
by which it reached this state in the original motion. If in
that motion H was decreasing, then obviously in this reversed
motion H will increase, and so to every state of motion in
which H decreases there corresponds another in which it
increases. The solution of this paradox is revealed when
attention is drawn to the fact that in the previous section
the various formulsD are values for the probable number of
collisions per unit time ; the predicted behaviour of the H
function is not given with absolute certainty ; only its most
probable but not perfectly certain behaviour is given by the
result. Presumably, if the gas is not in a state of equili-
brium, H will in all likelihood decrease ultimately to its
minimum equilibrium value, but it is not guaranteed that it
may not in the meantime fluctuate now and then to higher
values than that possessed at the moment. In fact, the
reader will " see the light " when he recalls the standpoint
of Statistical Mechanics as he has learnt it in the text,
especially in Chapters XXIII. and XXIV, We can make no
definite prediction about the behaviour of the system in any
given state ; the conditions are too complex to work out in
detail. We can say with some assurance what is its most
likely behaviour, our ideas of probability being based on the
general behaviour of an ensemble of similar systems. Indeed,
we have in effect met the H function quite early in the book.
If, instead of using the integral notation of (II. 2), we revert
to our original notation in connection with the partition of
a velocity-diagram into c cells, then for a distribution in
APPENDIX ON COLLISION-FORMULAE, ETC. 311
which there are n l representative points in the first cell,
n a in the second, etc., the H function is essentially
r-l
apart from a constant term involving the size of the cell.
Thus H is just
n log n W (n v ?i 2 , . . . , n c )
and the conclusions drawn previously concerning the
probable but not certain increase of the W function to a
maximum value are just as true in this strictly limited sense
concerning the behaviour of the H function. It was, as we
have stated, in this clarification of the H theorem that the
true standpoint of Statistical Mechanics, freed, as far as
equilibrium conditions are concerned, from all connection
with special collisional mechanisms, stood revealed.
III. The Kinetics of Gas Reactions in a Homogeneous
System. If in a mixture of two gases a reaction takes place
in which the two dissimilar molecules unite to form a new
molecule or exchange parts to yield two different molecules,
it is clear that the process is dependent on collisions in some
way. If, however, a reaction occurs in a simple gas in
which the original molecules decompose into two or more
molecules, it is not so obvious that collisions necessarily play
any part in the process.
Taking the first case, there is very clear evidence that the
rate of the reaction, where this is slow enough to be measured,
is not simply dependent on the collision-frequency, i.e.,
simply proportional to it. Of course, there does not exist a
state of equilibrium if a chemical reaction is going on ;
nevertheless, the formulae of Section I. for the equilibrium
state will apply approximately if the rate is slow enough,
and if the rate were proportional to the total number of
collisions per second between the two types of reacting
molecules, then the rate would be equal to
KC.C,
where K is the so-called " velocity-constant " and C l and <7 a
are the concentrations of the molecules. Now, undoubtedly,
quite a number of gas reactions occur in which the rate is
312 STATISTICAL MECHANICS FOR STUDENTS
proportional to the product of the concentrations ; and
there exist simple gas reactions in which the rate is pro-
portional to the square of the concentrations. Yet a glance
at (I. 3) or (I. 4) will show that on such an assumption the
velocity-constant would vary as 6* where 6 is the absolute
temperature at which the reaction is allowed to take place ;
but this is violently at variance with the facts ; for in such
reactions of the second order as have been carefully studied
the value of K will double or even treble for a rise of tempera-
ture comparable with 10.
As every physical chemist knows, it was Arrhenius who
was the fiist to offer a hypothesis to deal with this dis-
crepancy. About 1890 he suggested that the reaction did
not take place between molecules in their normal state, but
between molecules in an " activated " state, i.e., that the
change was due to collisions between special groups of
molecules, and this hypothesis fits very nicely into the
" quantum scheme of things " now existing. For whatever
might be the change that is produced in the structure of a
molecule by " activation," it was postulated that it was
effected by the acquisition of energy far above the normal
amount in an average molecule, so that nowadays we simply
regard an activated molecule as a molecule in a higher
quantum state. For a full discussion of the idea of activa-
tion, we must refer the reader to a text of Physical Chemistry.
(See, for instance, W. C. Lewis's System of Physical
CJiemistnj, Vol. I., Chapter IX. ; Vol. III., Chapter VII.)
All that we are concerned with at the moment is to show
how on statistical grounds it leads very simply to Arrhenius'
well-known law of the change of velocity constant with
temperature. It is quite sufficient for our purpose to assume
that each molecule has, as regards internal phases, a normal
or lower quantum state indicated by the suffix n, and one
upper quantum state (activated) indicated by the suffix a.
The result is just as easily proved for the assumption of many
higher quantum states, but the mathematical expressions are
more complicated to handle.
Thus, for the molecules of type 1, the number in the lower
quantum state are proportional to w ln exp ( p e lw ) where
APPENDIX ON COLLISION-FORMULA, ETC. 313
w ln is the a priori probability of that state and e ln the
internal energy. The number in the activated state is
proportional to w la exp ( p e la ). (Observe that we are
assuming the expressions for the equilibrium state to be
sufficiently good approximations for a state of reaction ;
this will require some consideration later.) For the second
molecule similar expressions hold. Thus the number of
activated molecules of type 1, existing at the moment
when there are altogether N l molecules of this type which
have not yet gone into reaction, is
W
la
The activated molecules of type 2 are
Arrhenius' assumption is that the rate of reaction is pro-
portional in the collision-frequency of the activated molecules.
It is not necessarily assumed that all such collisions lead to a
reaction ; other circumstances, such as state of orientation,
for example, might have to be taken into account ; but it
is assumed that a definite fraction of such collisions are
effective. Calling this fraction / 12 , we see that the rate of
reaction taken to be the rate at which molecules pass out of
the reactant condition, viz., -~ d NJdt ord N 2 /dt, is obtained
as the product of (III. 1), (III. 2),/ 12 and 2 a 2 (27r//zm 12 )*.
It is further assumed that / 12 does not depend on the
temperature, being a purely molecular property. Writing
*_N 1 _dNt_
dt & "*"*
we can easily obtain the expression for K, and on taking the
logarithm we find it to be equal to the sum of the following
seven expressions
log {2a 2 27r/&m 12 }
, t - a
log (w ln exp ( ij, e lH ) + w la exp (- p e la )[
- log [w 2n exp (- /* ta ) + w 2n exp (- /* e 2n )}
314 STATISTICAL MECHANICS FOR STUDENTS
If we proceed to differentiate log K with respect to 0, we
find that the first two expressions above contribute nothing,
and d log K/dO is equal to the sum of five expressions
1
2~0
la
W
ln
_
k e*{w ln exp(- p, e ln ) + w la exp (- p lfl )}
and a similar expression for type 2 molecules.
A little thought will show that the fourth of these is
simply lm /k 2 , where e lm is the average energy of all the
molecules of type 1. In effect, e lm is but little different
from c ln , if the latter is much less than e la .
To sum up, we find that
d e
where is written for (t la + e 2a ) (e lm + e 2m ), i.e., the
amount by which the combined internal energies of the
activated molecules exceeds their combined energies in the
average state, or practically in the lower quantum un-
activated state. This clearly corresponds to Arrhenius'
" energy of activation," or as it is sometimes called the
" critical increment " of energy. Practically %k 6 is
insignificant compared to , and we obtain Arrhenius' well-
known equation
d log K:
/7 ft If #2 *
\JU \J K U
d^_ J_
C/JL 7 /\ " 1C j f\n
As is easily shown, this is quite consistent with the very
rapid increase of K with temperature. Indeed (III. 3) is
APPENDIX ON COLLISION-FORMULA, ETC. 315
used in connection with the experimental determinations of K
over wide ranges of temperature to determine the value of .
For details the reader is once more referred to texts of
Physical Chemistry.
Despite the rather clumsy appearance of the mathematical
expressions involved, and the still more complicated expres-
sions employed if we had used several upper quantum states
instead of one, the Arrhenius result is essentially dependent
on the factors of the type e~ M in the formulae for the numbers
of activated molecules, and it is apparently very satisfactory
that the result should follow so directly from this universal
characteristic of statistical formulae. But there is a very
important feature of the treatment which has to be dealt
with, and which has in certain cases presented very serious
difficulties.
The reaction removes the activated molecules from the
original system ; they form molecules of the resultant
substances in an upper quantum state ; their excess energy
is lost in subsequent collisions (presumably), and they
become resultants in a normal state, the algebraic difference
between the energy of activation and the excess energy of
resultants subsequently lost being the ordinary heat of
reaction (positive or negative) per molecular group. Of
course, we know that among the remaining unactivated
molecules a redistribution of internal energies would take
place, leading once more to the usual statistical arrangement,
provided the chemical reaction did not go on. It is clear,
therefore, that in order to use equilibrium formulae, even for
approximate calculations, we must be satisfied that the
reaction goes so slowly that there is always in existence a
group of activated molecules not too small in number
compared with the equilibrium number, and that implies
that the rate of production of activated molecules is at least
equal to the rate at which they are removed by the reaction.
Here, then, we are faced with a problem of mechanism.
We know that the system when denuded of activated
molecules will proceed to make them good. But will it
proceed at a fast enough rate? Mechanisms differ, as we
pointed out above, not in their ultimate goal, but in the rate
316 STATISTICAL MECHANICS FOR STUDENTS
at which they reach it. Let us consider the mechanism of
simple collisions in this connection. On p. 300 there is
quoted a well-known formula for the combined kinetic
energies of translation of two molecules approaching a
collision. If the collision were such as to destroy the
relative motion so that the molecules travelled thereafter
with a common velocity (the collision being thus completely
inelastic) that velocity would by the principle of conserva-
tion of momentum be c, and the kinetic energy would be
| (m l + m 2 ) c 2 . Thus \ m 12 r 2 , the " relative kinetic energy "
before collision, would be converted to some extent into
internal energies of the molecules ; we say " to some
extent," for part of it might appear as an energy of rotation
of the combined molecules. The point is that even in this
case of a collision without rebound, ^ ra 12 ?' 2 gives the upper
limit of original kinetic energy available for subsequent
internal energy, i.e., for activation ; and in other collisions
even less would be available. If, therefore, we consider the
rate at which molecules with a definite lower limit of relative
kinetic energy come into collision with one another, we have
some means of estimating if the collision-mechanism is
suitable for a sufficient rapid supply of activated molecules
whose energy of activation is equal to the assigned lower
limit. The necessary formulae have been worked out in
Section I.
Putting for the energy of activation, formula (I. 8)
shows us that the number of collisions with a relative
kinetic energy greater than is
-^) .... (III. 4)
Hinshelwood's book, The Kinetics of Chemical Change
in Gaseous Systems, provides a number of experimental
results to test this formula. The decomposition of hydrogen
iodide is a familiar example of a second order reaction, and
we assume for the moment that the activation is due to
collisions of normal molecules of HI and the decomposition
into H 2 and I 2 due to collisions of activated molecules. The
heat of activation per gram-molecule is 44,000 calories, and,
taking the gas constant per gram molecule as 2 calories per
APPENDIX ON COLLISION-FORMULAE, ETC. 317
degree, this gives 22,000 as the value for /&. The reaction
has been studied over the range 550 K to 780 K ; so,
putting as approximately 600, we see that /* (i.e., /&0) is
over 30, and in comparison unity is negligible in the second
factor of (III. 4). The formula for C is given in (I. 4), and
so the rate of collision for molecules with relative kinetic
energy above is practically
.... (HI. 5)
In this we can put a = 2 x 10~ 8 , m = 210 X 10~ 24 ,
lc = 1-35 X 10' 16 , /& = 22,000, 9 = 600, N = 6 X 10 20
(i.e., a concentration of 1 gram-molecule per litre). The
result is approximately
2 X 10 14
On the other hand, the observed rate of reaction shows
that about
2 X 10 13
molecules of HI (at a concentration of 1 gram-molecule per
litre and a temperature, 600 K) react per second. So there
would appear to be a " factor of safety " of about 10, which
does not appear to be too much to spare, since only a
favourably circumstanced fraction of the 2 x 10 14 collisions
per second can result in activation. But we must not over-
look one assumption which we are implicitly making at the
moment. It is that all the spare kinetic energy is going
into one of the colliding molecules, or that about one-tenth
of all the collisions worth considering result each in one
activated molecule. Indeed, the treatment so far is tanta-
mount to assuming that one-tenth of a certain group of
suitable collisions result at once in the reaction ; for a little
thought will show that the intermediate state of activation
and subsequent collision between two activated molecules,
each one arising from a different collision, is an unnecessary
part of the picture as we have been treating it so far. In-
deed, if we consider such a simple collision theory of reaction
in which activation is not required (or at most is merged
318 STATISTICAL MECHANICS FOR STUDENTS
into the process of collision), we can write for the rate of
reaction
N* xa* ( ~V e~* expert) (HI- 8)
where a is an average value for the fraction of collisions
which result in reaction. As matters stand now, we would
have to take about 1/10 for the value of a in the case of
hydrogen iodide, and, in other cases, the fraction does not
appear to be any less. However, there are other considera-
tions bearing on internal energy which tend to show this
simple collision theory of reaction in a more favourable light.
Before considering these, however, let us glance for a
moment at a more genuine activation theory, i.e., one in
which collisions leading to activation and collisions of acti-
vated molecules leading to reaction are distinct occurrences.
Looking back to our earlier expressions, we see that the
energy of activation is the sum of e la e lm and e 2a e 2w ,
so that it is not necessary to assume that an activating
collision must give to either of the molecules the whole of
the energy of activation. In the case of hydrogen iodide,
for instance, an activating collision is only required to give
one of the molecules engaging in it half the erergy of activa-
tion ; in a subsequent collision with another molecule
activated with the half amount, the total amount is then
presumably available for the splitting into H 2 and I 2 . This
means that in using (III. 5) for the calculation of the upper
limit of the number of collisions which might result in
activation, we do not write exp ( 22,000/0) for exp ( //,),
but exp ( 11,000/0), which is practically 18 times as great,
and this certainly offers an ample margin for favourable
collisions.
Thus, while there is a good deal to be said on statistical
grounds for a theory which would consider that bimolecular
reactions occur through the collisions of previously activated
molecules, there must be an element of doubt about a simple
collision theory in which we would assume that normal
molecules can react at once if they collide with sufficient
relative velocity. Indeed, if we calculate, as Tolman does,
on a more stringent basis, which assumes that the kinetic
APPENDIX ON COLLISION-FORMULA, ETC. 319
energy corresponding to the resolved component of the
relative velocity along the line of centres is the only available
source for the reaction, then even the factor of safety of 10
or thereabouts disappears, and we have, in general, just a
rough equality between rate of reaction and rate of favour-
able collisions.
However, on other grounds, the theory of activation does
not seem to be in such favour nowadays among physical
chemists, and R. N. Fowler has pointed out that the simple
collision theory can be rendered much more plausible by
introducing available internal energy considerations, as well
as merely available relative kinetic energy. In deriving
Arrhenius' expression, we assumed for simplicity of writing
just two quantum states, a normal and an activated ; but it
is much more probable that in reality there are a number of
quantum states between the normal and the activated.
Hence, in considering collisions with a definite relative
kinetic energy, we can assume that some of these will be
between molecules whose internal states, while not high
enough for activation, are not as low as the normal. These
would contain internal energy available for reaction pur-
poses, and would obviously allow us to put the necessary
amount of relative kinetic energy at a lower figure than
before, and, in effect, bring into the favourable field many
collisions previously ruled out. To render the necessary
mathematical analysis which develops this idea as simple as
possible, we will revert to the picture of internal harmonic
oscillators as the seat of this internal energy, and carry out
the calculations along classical lines. The internal energy is
(for / oscillators)
If we wish to calculate the number of molecules whose
internal energy lies between 77 and 77 + 877, we must integrate
for all values of q l . . . . p f which correspond to
320 STATISTICAL MECHANICS FOR STUDENTS
divide the result by the same integral over all values of
q v . . . , p fy and multiply by N. The method of carrying
out such integrations as occur in the numerator is associated
with the name of Dirichlet, and will be found in standard
texts of mathematical analysis.* The result sought for is
known to be
f 00
1 x f ~ l exp ( JJLX) dx
Jo
The denominator is, of course, equal to
1 f 00
~ y f ~ 1 exp(y)dy
PJo
and the integral is known to be equal to (/ 1) !
Hence the number of molecules whose internal energy lies
between 77 and 77 -f- 877 is
NfL/ r-i-n (in. 7)
(/-I)! '
Using (I. 7) and (III. 7), we can now calculate the number
of collisions per unit volume per unit time between molecules
of type 1 with internal energy between 77 1 and 77 x + 877!, and
molecules of type 2 with internal energy between 772 and
772 + 8772, the relative kinetic energy being between and
| -f- S. It is the product of
2 a* -L exp (-
\ >IZ /
and two expressions of the form (III. 7). The result is
[ p. (rj l + rj 2 + f )j 87) v 8773, 8^
To calculate the number of collisions for which the energy
available for reaction purposes, viz., ^1 + ^2 + >i g greater
than an assigned value we must integrate this expression for
* Whittaker and Watson's Modern Analysis (3rd Edition), p. 238.
APPENDIX ON COLLISION-FORMULAS, ETC. 321
all values of rj l9 7? 2 , , satisfying this condition. Dirichlet's
method is once more used, and yields the result
ao^y. / 2. V /1+/1+ ,C I+/t+1< .- Mrfr
(/!+/+ 1) I \ii/ J
which is equal to
(A+A+i)!
If /*, or /kd is sufficiently large, this is equal to
(A
If the molecules are of one type the factor 2 must as usual
be removed, and we obtain
n2/+ 1
- exp ( //,) . (III. 8)
Let us assume then that (III. 8) gives the law of reaction
, where
^.
1)1
a being some fraction giving on the average the relative
number of sufficiently energetic collisions which result in a
reaction ; then, taking the logarithm of K and differentiating,
we find
dlogK _ _2/+|
d 9 k9* 6
But, experimentally, we find the energy of activation
from the Arrhenius formula
dlogK __ ^
d0 ~~lc6* 9
where e indicates the experimental value of the energy of
activation as distinct from , which represents the lower
limit of the theoretical available energy.
322 STATISTICAL MECHANICS FOR STUDENTS
Hence
and so (III. 8) becomes, in terms of the experimentally
determined energy of activation,
while (III. 5) becomes
which is just (III. 9) when / is made zero.
Now (III. 9) bears to (III. 10) a ratio which is practically
(2/+1)!
an expression which may easily be greater than unity.
Thus, for hydrogen iodide, / is at least unity, and the ratio
becomes if that value is assumed
<r 2 .
As e is more than thirty times as great as kO in this case,
this ratio is about 20, and it follows that to meet the experi-
mental rate of reaction, the simple collision theory, if
supplemented by this hypothesis of internal available
energy, would only require a to have on the average a value
about 1/200, which leaves a very good margin indeed.
On statistical grounds, then, the collision theory of bi-
molecular homogeneous gas reactions can be regarded in a
favourable light. It is, however, a well-known fact that
serious difficulty has hitherto attended similar considerations
when applied to reactions of the first order in which the rate
of reaction is proportional to the first power of concentration
and not to its square, the best discussed example being the
decomposition of nitrogen pentoxide. We can, of course,
as before, assume that the molecules entering into reaction
have been previously activated by collision, but it is hard to
admit that reaction is duQ to collision of activated molecules,
since the rate of reaction would surely be proportional to
APPENDIX ON COLLISION-FORMULAE, ETC. 323
square of concentration and not to first power. In order to
evade this difficulty, it is generally assumed that once a
molecule has been activated, there is a definite chance that
before it can be deactivated by another collision, it will
spontaneously disintegrate. That is, collisions leading now
to activation and now to deactivation, maintain an equili-
brium distribution in the usual statistical manner between
the normal and various quantum states (including the
activated state) or nearly so, for combined with this is this
postulated disintegration mechanism which tends to upset
this distribution (but not too rapidly) by denuding the
system of its activated molecules. Since the concentration
of activated molecules is proportional to exp ( juej, it
follows that the number of activated molecules at any
moment is
w n exp (- ii n ) + w a exp (- pc a )
and remains practically unaffected if the rate of reaction is
slow enough ; as there is a definite chance that such a
molecule will disintegrate before another collision tends to
deactivate it, this rate of reaction will be proportional to the
expression just written and so to the first power of N.
Moreover, K will be equal to
w n exp(p, e n ) + w a exp(p, fl )
where A is a constant independent of the temperature. Taking
the logarithm and differentiating with respect to 0, we find
just as before Arrhenius' equation
d log K
dO = kid 2
where = e a e m , or practically a e n .
This picture of the occurrence is due to Lindemann
(Trans. Faraday Soc. y Vol. 17, p. 599 [1921]). The difficulties
attending it have as usual centred round the necessity of
finding the rate of activation to be fast enough. It is im-
possible in this short appendix to go into the matter at any
length. The case of nitrogen pentoxide proves to be the
most refractory. Thus if activation had only relative
Y2
324 STATISTICAL MECHANICS FOE STUDENTS
movement to look to for its energy, it can be shown that it
would take place at a rate about -0001 of that required.
That is Tolman's view, using his very stringent formula
involving relative motion resolved along the line of centres
as the only effective source of energy of activation. (See
Tolman's Statistical Mechanics, Chapter XXI., section 326.)
To be sure, if the full relative motion is considered, matters
do not look so bad, and, as Fowler shows, if we supplement
the energy of relative motion with internal energy, using a
reasonable number of internal degrees of freedom, we can
just bring the case of nitrogen pentoxide within the bounds
of possibility without anything to spare, however ; more-
over, there are three or four other unimolecular gaseous
reactions which are well within the grasp of Lindemann's
hypothesis, as amplified by Fowler, without making too
great demands on the probabilities of the situation. (For
details, consult Fowler's Statistical Mechanics, Chapter XVII.
section 4.)
Apart from Fowler's considerations, however, and indeed
some years before he advanced them, Christiansen and
Kramers (Zeitschrift fur physik. Chem., Vol. 104, p. 451
[1923] ) attempted to meet the difficulty of Lindemann's
hypothesis by pointing out that the resultants of the decom-
position would contain energy above the normal ; such
energy must leave them when becoming normal resultant
molecules, and it seems natural to look for the cause of this
loss in ordinary collisions which convert this energy into
heat motion of the molecules. But these authors suggest
that this energy might be used at all events in part for
activation purposes. In short, activated molecules of the
resultants will collide with unactivated molecules of the
reactant and transfer this energy as internal energy to the
latter, little or none of it transforming into translatory
kinetic energy. The difficulties attending this hypothesis
are discussed by Tolman in the reference cited above.
Finally, it remains to refer briefly to another theory of
activation propounded nearly fifteen years ago by W. C.
Lewis, and independently about the same time by Perrin.
A full account of it will be found in Lewis's System of Physical
APPENDIX ON COLLISION-FORMULAE, ETC. 325
Chemistry, Vol. III., Chapter VII. It looks for the energy
of activation, not to collisions but to the thermal radiation
surrounding the reacting molecules, and in temperature
equilibrium with the walls of the enclosing vessel. It
assumes that a molecule to become active absorbs a quantum
of energy whose frequency corresponds to the value given
by the equation, energy of activation = hv. If one uses the
expressions for rate of absorption of radiation developed in
the Planck theory of full radiation, however, this hypothesis
finds itself in as bad, if not a worse, situation as regards
nitrogen pentoxide than the simple collision theory of
activation. The advantage of it is that it gives a simple
explanation of the unimolecular nature of the reaction
without resort to hypotheses of spontaneous disintegration.
As regards bimolecular reactions, the radiation theory is
quite satisfactory if it is assumed that the radiation has only
to supply a portion (say one-half) of necessary energy of
activation to one molecule and the remaining portion to the
other, collisions between previously activated molecules
producing the reaction. This radiation theory seems to
have receded into the background lately, although Tolman
in his book discusses it in a very favourable way, showing
how an elaboration of it may be possibly made to fit the
facts. The author had many private discussions on this
matter with Professor Lewis some years ago, and has
attempted to present the simple radiation theory in a form
which yields a criterion distinguishing molecules engaging
in unimolecular reactions from those engaging in bimolecular
and at the same time suggests a possible escape from the
rate of activation difficulty (Rice, " Note on the Radiation
Theory of Chemical Reaction/ 7 Trans. Faraday Soc., No. 63,
Vol. XXI., Part 3). Any reader with a sufficient knowledge
of electromagnetic theory could follow the argument
advanced in this note, but the author must point out that
his reasoning leads to an abnormal reflecting power of
nitrogen peroxide for radiation of the activating frequency
which would be just below the visible red, and some un-
published observations carried out by E. A. Stewardson at
the author's request do not confirm this.
326 STATISTICAL MECHANICS FOR STUDENTS
As matters stand now, it would appear that a collision
mechanism is the least unsatisfactory hypothesis for activa-
tion. It is quite possible, in view of the solution of similar
difficulties of quantum states in other processes by the New
Mechanics, that the theory of activation, if it survives, will
look in this direction for a better formulation.
As regards the application of the formulae of Section I. to
the treatment of viscosity and diffusion in gases, the reader
will find a sufficiently elementary account in the English
translation of Bloch's Kinetic Theory of Gases (published by
Methuen), and in view of the elegant and simple treatment
to be found in Chapter III. of that little volume, there
appears to be no need to continue this appendix further.
NOTE ON CHAPTER X
The Smoluchowski formula used in connection with
fluctuations was derived in Chapter X. from a rather detailed
consideration of the circumstances, so as to make the argu-
ment as concrete as possible to the beginner. It can,
however, be derived in a more abstract fashion by a method
due to Einstein.
Consider the normal state of a system in which the
internal energy is U and the free energy F, so that
F== U - 6S
where S is the entropy, or
U-F
Indicate a state to which the system can fluctuate by
primed symbols, the internal energy and temperature still
being the same however. Let this state be one to which
the system could be brought from the normal by external
work of amount S E performed on the system. Then since
8F = -SS 6+ SE
and S is zero, it follows that
F' - F + 3 E
U-F-SE
and S =
SE
But 8 = k\ogW
and S' = k log W
327
328 STATISTICAL MECHANICS FOR STUDENTS
by Boltzmann's formula. Hence
W _ fS' - S
W =
or W = W exp
which is Einstein's general formula. In Chapter X.
8E = - f" (p - p ) dv
J v
and Smoluchowski's result follows.
SUGGESTIONS FOR FURTHER
READING
THOSE readers who wish to pursue the subject further
mainly for its applications in Physics and Chemistry will
find an excellent guide in R. C. Tolman's " Statistical
Mechanics " (American Chemical Society Monograph Series,
published by the Chemical Catalog Company, New York).
The recently published " Statistical Mechanics " of R. H.
Fowler is a very exhaustive treatise and, for those who
possess sufficient mathematical equipment, a veritable mine
of information on the many topics, physical, chemical and
astrophysical, to which the statistical method can be applied.
It is published by the Cambridge University Press. These
books give numerous references to the original literature.
The classical work is, of course, Willard Gibbs' " Elemen-
tary Principles in Statistical Mechanics " (Yale University
Press). Based on the treatment of ensembles of similar
dynamical systems its " modest aim," in the words of its
author, is that " of deducing the more obvious propositions
relating to the statistical branch of mechanics." It does
not in the main concern itself with thermodynamic phe-
nomena or with the " mysteries of Nature," confining itself
to logical deduction without reference to hypotheses con-
cerning the constitution of matter. Nevertheless, no serious
student of the subject should fail to read it, and its four last
chapters do at all events discuss the relations of the subject
to natural phenomena with that insight for which the
author was so justly famous.
A very important monograph by P. and T. Ehrenfest
discusses the origin of the subject and the gradual clarifica-
tion of its fundamental postulates. Its title is " Begriffliche
Grundlagen der statistischen Auffassung in der Mechanik."
329
330 SUGGESTIONS FOR FURTHER READING
It is Heft 6 of Band IV. 2 II. of the " Encyclopadie der
Mathematischen Wissenschaften."
There are several works on the applications of the statis-
tical method to gases. The book for those whose time and
mathematical knowledge are limited is E. Bloch's " Kinetic
Theory of Gases," a translation of which is published by
Methuen. The standard works are, of course, Boltzmann's
" Vorlesungen iiber Gastheorie," and Jeans' " Dynamical
Theory of Gases." An American work, " Kinetic Theory of
Gases," by Loeb (Published by Ginn & Co.), combines the
usual theoretical treatment with useful accounts of recent
experimental research on gases.
INDEX
Action, 79
Action-integrals, 140
A priori probability, 6
Avogadro's hypothesis, 51
Bohr's postulates, 147
Bose's statistics of light-quanta, 270
Characteristic temperatures, 210
Chemical constant, 215
Chemical kinetics, 311
Clapeyroii's equation, 120
Collision-frequency, 296
Complexions of a molecular system, 21, 24
Condensation, 104
Contour integration applied to statistical calculations, 284
Darwin and Fowler's statistical method, 282
Debye's theory of specific heats of solids, 209
Degeneracy in a conditionally periodic system, 158, 161
Distribution constant, 47, 49
, modulus of, 263
Einstein's fluctuation formula, 327
statistics of an ideal gas, 275
theory of specific heat of solids, 207
Elastic spectrum, 175, 189
Energy, average, 45
equipartitioti of, 57, 60, 266
hypersurfaces, 153, 156
of vibrating lattice, 181, 195
Ensemble, canonical, 261
micro canonical, 263
of systems, 234, 246
Entropy, 69
constant, 80, 213, 221, 258, 279
kinetic, 75
of a perfect gas, 77
Equation of state of a perfect gas, 47
Equilibrium, chemical, 82
Error, mean square, 19
Errors, normal law of, 11, 14
331
332 INDEX
Fermi-Dirac statistics, 277
Fluctuations, 102
Fourier's theorem, 175, 176
Free -energy, 75
Full radiation, 205, 275
Gas reactions, 82
Gibbs' canonical ensemble, 261
microcanonical ensemble, 263
phase-space, 254
thermodynamic analogies, 266
Hamilton's equations, 246
H-theorem, 306
Intermolecular forces, 92
Lattice, cubical, 123, 189
energy of, 195
vibration of, 192
Lattice, linear, 175
energy of, 181
vibration of, 176
Lattice, superficial, 186
Lattices, cubical, statistics of a system of, 198
Lattices, linear, statistics of a system of, 183
Light-quantum, 141, 270
Liouville's theorem, 248
Maxwell-Boltzmann law, 43, 277
Mean-squared velocity, 56
Microscopic states, 252
Mixture of gases, 49
Modulus of distribution, 263
Molecular phase-space, 254
Molecules, finite size of, 92
Nernst's heat theorem, 213
Normal state of a system, 68
Nul -point energy, 141, 281
Oscillator, 57, 131
Parameters of a system, 38
Partition functions of Darwin and Fowler, 282, 290
Pauli's exclusion principle, Fermi's adaptation of, 277
Phases, 24
Phase-cell, 25, 77
finite magnitude of, 136, 221
Phase diagram, 26
Planck's constant, 80, 130, 134
law of distribution, 134, 137, 141, 156, 273, 293
for full radiation, 205, 275
INDEX 333
Planck-oscillator, 131
Pressure of a fluid, 53
, intrinsic or internal, 54, 94, 96
Pressures, law of partial, 62
Probability, d priori, 6, 34, 151
and intermolecular action, 97
state of maximum, 39
Propagation of a disturbance in an elastic solid, 190
Quantisation of paths, 134, 147, 158
Quantum hypothesis, 126
states, 136, 138
Rayleigh-Jeans law for full radiation, 205
Rotational specific heat, 166
Saturated vapour, theory of, 118
Second law of thermodynamics, 68, 114
Smoluchowski's theory of fluctuations, 106
Solid state, 123
Specific heat, 65, 123, 164, 166, 277
Standing waves in a lattice, 200
Stationary states, 143
Steepest descents, method of, 284
Stern's treatment of the entropy-constant problem, 226
Stirling's theorem, 24, 28
Temperature, and distribution constant, 45
characteristic, 210
Thermodynamic equilibrium, 68
Trajectory, 26
Unstable homogeneous states of a fluid, 96
Van der Waal's equation, 95, 101
Vapour -pressure constant, 215
Wave -function, 189
PRINTED IN GREAT BRITAIN BY THE WHITEFRIARS PRESS, LTD.,
LONDON AND TONBRIDQE.
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11